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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempntlemh 20801* Lemma for pnt 20816. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   =>    |-  ( ( ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J )  /\  ( K ^ J )  <_  ( sqr `  Z )
 ) )
 
Theorempntlemn 20802* Lemma for pnt 20816. The "naive" base bound, which we will slightly improve. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   =>    |-  ( ( ph  /\  ( J  e.  NN  /\  J  <_  ( Z  /  Y ) ) )  -> 
 0  <_  ( (
 ( U  /  J )  -  ( abs `  (
 ( R `  ( Z  /  J ) ) 
 /  Z ) ) )  x.  ( log `  J ) ) )
 
Theorempntlemq 20803* Lemma for pntlemj 20805. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   &    |-  ( ph  ->  V  e.  RR+ )   &    |-  ( ph  ->  ( ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E ) )  x.  V )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  J  e.  ( M..^ N ) )   &    |-  I  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  V ) ) )   =>    |-  ( ph  ->  I  C_  O )
 
Theorempntlemr 20804* Lemma for pntlemj 20805. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   &    |-  ( ph  ->  V  e.  RR+ )   &    |-  ( ph  ->  ( ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E ) )  x.  V )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  J  e.  ( M..^ N ) )   &    |-  I  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  V ) ) )   =>    |-  ( ph  ->  (
 ( U  -  E )  x.  ( ( ( L  x.  E ) 
 /  8 )  x.  ( log `  Z ) ) )  <_  ( ( # `  I
 )  x.  ( ( U  -  E )  x.  ( ( log `  ( Z  /  V ) )  /  ( Z  /  V ) ) ) ) )
 
Theorempntlemj 20805* Lemma for pnt 20816. The induction step. Using pntibnd 20795, we find an interval in  K ^ J ... K ^ ( J  + 
1 ) which is sufficiently large and has a much smaller value,  R ( z )  / 
z  <_  E (instead of our original bound 
R ( z )  /  z  <_  U). (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   &    |-  ( ph  ->  V  e.  RR+ )   &    |-  ( ph  ->  ( ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E ) )  x.  V )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  J  e.  ( M..^ N ) )   &    |-  I  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  V ) ) )   =>    |-  ( ph  ->  (
 ( U  -  E )  x.  ( ( ( L  x.  E ) 
 /  8 )  x.  ( log `  Z ) ) )  <_  sum_ n  e.  O  ( ( ( U  /  n )  -  ( abs `  ( ( R `
  ( Z  /  n ) )  /  Z ) ) )  x.  ( log `  n ) ) )
 
Theorempntlemi 20806* Lemma for pnt 20816. Eliminate some assumptions from pntlemj 20805. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  O  =  ( ( ( |_ `  ( Z  /  ( K ^
 ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z 
 /  ( K ^ J ) ) ) )   =>    |-  ( ( ph  /\  J  e.  ( M..^ N ) )  ->  ( ( U  -  E )  x.  ( ( ( L  x.  E )  / 
 8 )  x.  ( log `  Z ) ) )  <_  sum_ n  e.  O  ( ( ( U  /  n )  -  ( abs `  (
 ( R `  ( Z  /  n ) ) 
 /  Z ) ) )  x.  ( log `  n ) ) )
 
Theorempntlemf 20807* Lemma for pnt 20816. Add up the pieces in pntlemi 20806 to get an estimate slightly better than the naive lower bound  0. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   =>    |-  ( ph  ->  (
 ( U  -  E )  x.  ( ( ( L  x.  ( E ^ 2 ) ) 
 /  (; 3 2  x.  B ) )  x.  (
 ( log `  Z ) ^ 2 ) ) )  <_  sum_ n  e.  ( 1 ... ( |_ `  ( Z  /  Y ) ) ) ( ( ( U 
 /  n )  -  ( abs `  ( ( R `  ( Z  /  n ) )  /  Z ) ) )  x.  ( log `  n ) ) )
 
Theorempntlemk 20808* Lemma for pnt 20816. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   =>    |-  ( ph  ->  (
 2  x.  sum_ n  e.  ( 1 ... ( |_ `  ( Z  /  Y ) ) ) ( ( U  /  n )  x.  ( log `  n ) ) )  <_  ( ( U  x.  ( ( log `  Z )  +  3 ) )  x.  ( log `  Z ) ) )
 
Theorempntlemo 20809* Lemma for pnt 20816. Combine all the estimates to establish a smaller eventual bound on  R ( Z )  /  Z. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  Z  e.  ( W [,)  +oo )
 )   &    |-  M  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )   &    |-  N  =  ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2
 ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   &    |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( K  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  A. z  e.  ( 1 (,)  +oo ) ( ( ( ( abs `  ( R `  z ) )  x.  ( log `  z
 ) )  -  (
 ( 2  /  ( log `  z ) )  x.  sum_ i  e.  (
 1 ... ( |_ `  (
 z  /  Y )
 ) ) ( ( abs `  ( R `  ( z  /  i
 ) ) )  x.  ( log `  i
 ) ) ) ) 
 /  z )  <_  C )   =>    |-  ( ph  ->  ( abs `  ( ( R `
  Z )  /  Z ) )  <_  ( U  -  ( F  x.  ( U ^
 3 ) ) ) )
 
Theorempntleme 20810* Lemma for pnt 20816. Package up pntlemo 20809 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  W  =  ( ( ( Y  +  ( 4  /  ( L  x.  E ) ) ) ^ 2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  ( (
 (; 3 2  x.  B )  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  (
 ( U  x.  3
 )  +  C ) ) ) ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  (
 ( R `  z
 )  /  z )
 )  <_  U )   &    |-  ( ph  ->  A. k  e.  ( K [,)  +oo ) A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  E ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )   &    |-  ( ph  ->  A. z  e.  ( 1 (,)  +oo ) ( ( ( ( abs `  ( R `  z ) )  x.  ( log `  z
 ) )  -  (
 ( 2  /  ( log `  z ) )  x.  sum_ i  e.  (
 1 ... ( |_ `  (
 z  /  Y )
 ) ) ( ( abs `  ( R `  ( z  /  i
 ) ) )  x.  ( log `  i
 ) ) ) ) 
 /  z )  <_  C )   =>    |-  ( ph  ->  E. w  e.  RR+  A. v  e.  ( w [,)  +oo ) ( abs `  ( ( R `  v )  /  v
 ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
 
Theorempntlem3 20811* Lemma for pnt 20816. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  A )   &    |-  T  =  { t  e.  ( 0 [,] A )  |  E. y  e.  RR+  A. z  e.  (
 y [,)  +oo ) ( abs `  ( ( R `  z )  /  z ) )  <_  t }   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ( ph  /\  u  e.  T ) 
 ->  ( u  -  ( C  x.  ( u ^
 3 ) ) )  e.  T )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  ~~> r  1 )
 
Theorempntlemp 20812* Lemma for pnt 20816. Wrapping up more quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  A. e  e.  ( 0 (,) 1 ) E. x  e.  RR+  A. k  e.  ( ( exp `  ( B  /  e ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  e ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  e
 ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  e ) )   &    |-  ( ph  ->  U  e.  RR+ )   &    |-  ( ph  ->  U 
 <_  A )   &    |-  E  =  ( U  /  D )   &    |-  K  =  ( exp `  ( B  /  E ) )   &    |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y ) )   &    |-  ( ph  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  ( ( R `  z )  /  z
 ) )  <_  U )   =>    |-  ( ph  ->  E. w  e.  RR+  A. v  e.  ( w [,)  +oo ) ( abs `  ( ( R `  v )  /  v
 ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
 
Theorempntleml 20813* Lemma for pnt 20816. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a
 ) )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  RR+  ( abs `  ( ( R `  x )  /  x ) )  <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )   &    |-  D  =  ( A  +  1 )   &    |-  F  =  ( ( 1  -  (
 1  /  D )
 )  x.  ( ( L  /  (; 3 2  x.  B ) )  /  ( D ^ 2 ) ) )   &    |-  ( ph  ->  A. e  e.  ( 0 (,) 1 ) E. x  e.  RR+  A. k  e.  ( ( exp `  ( B  /  e ) ) [,)  +oo ) A. y  e.  ( x (,)  +oo ) E. z  e.  RR+  ( ( y  < 
 z  /\  ( (
 1  +  ( L  x.  e ) )  x.  z )  < 
 ( k  x.  y
 ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  e
 ) )  x.  z
 ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  e ) )   =>    |-  ( ph  ->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  ~~> r  1 )
 
Theorempnt3 20814 The Prime Number Theorem, version 3: the second Chebyshev function tends asymptotically to  x. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  ~~> r  1
 
Theorempnt2 20815 The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to  x. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  ( x  e.  RR+  |->  ( ( theta `  x )  /  x ) )  ~~> r  1
 
Theorempnt 20816 The Prime Number Theorem: the number of prime numbers less than  x tends asymptotically to  x  /  log (
x ) as  x goes to infinity. (Contributed by Mario Carneiro, 1-Jun-2016.)
 |-  ( x  e.  (
 1 (,)  +oo )  |->  ( (π `  x )  /  ( x  /  ( log `  x ) ) ) )  ~~> r  1
 
13.4.13  Ostrowski's theorem
 
Theoremabvcxp 20817* Raising an absolute value to a power less than one yields another absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  G  =  ( x  e.  B  |->  ( ( F `  x )  ^ c  S ) )   =>    |-  ( ( F  e.  A  /\  S  e.  (
 0 (,] 1 ) ) 
 ->  G  e.  A )
 
Theorempadicfval 20818* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( P  e.  Prime  ->  ( J `  P )  =  ( x  e. 
 QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) ) ) )
 
Theorempadicval 20819* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  X  e.  QQ )  ->  ( ( J `
  P ) `  X )  =  if ( X  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  X ) ) ) )
 
Theoremostth2lem1 20820* Lemma for ostth2 20839, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 20839. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, 
n  e.  o ( A ^ n ) for any 
1  <  A. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  (
 ( ph  /\  n  e. 
 NN )  ->  ( A ^ n )  <_  ( n  x.  B ) )   =>    |-  ( ph  ->  A  <_  1 )
 
Theoremqrngbas 20821 The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |- 
 QQ  =  ( Base `  Q )
 
Theoremqdrng 20822 The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  Q  e.  DivRing
 
Theoremqrng0 20823 The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  0  =  ( 0g
 `  Q )
 
Theoremqrng1 20824 The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  1  =  ( 1r
 `  Q )
 
Theoremqrngneg 20825 The additive inverse in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  ( X  e.  QQ  ->  ( ( inv g `  Q ) `  X )  =  -u X )
 
Theoremqrngdiv 20826 The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  ( ( X  e.  QQ  /\  Y  e.  QQ  /\  Y  =/=  0 ) 
 ->  ( X (/r `  Q ) Y )  =  ( X  /  Y ) )
 
Theoremqabvle 20827 By using induction on  N, we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( ( F  e.  A  /\  N  e.  NN0 )  ->  ( F `  N )  <_  N )
 
Theoremqabvexp 20828 Induct the product rule abvmul 15643 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( ( F  e.  A  /\  M  e.  QQ  /\  N  e.  NN0 )  ->  ( F `  ( M ^ N ) )  =  ( ( F `
  M ) ^ N ) )
 
Theoremostthlem1 20829* Lemma for ostth 20841. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ( ph  /\  n  e.  ( ZZ>= `  2 )
 )  ->  ( F `  n )  =  ( G `  n ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremostthlem2 20830* Lemma for ostth 20841. Refine ostthlem1 20829 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ( ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremqabsabv 20831 The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( abs  |`  QQ )  e.  A
 
Theorempadicabv 20832* The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  F  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( N ^ ( P  pCnt  x ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  N  e.  (
 0 (,) 1 ) ) 
 ->  F  e.  A )
 
Theorempadicabvf 20833* The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  J : Prime --> A
 
Theorempadicabvcxp 20834* All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( y  e. 
 QQ  |->  ( ( ( J `  P ) `
  y )  ^ c  R ) )  e.  A )
 
Theoremostth1 20835* - Lemma for ostth 20841: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If  F is equal to  1 on the primes, then by complete induction and the multiplicative property abvmul 15643 of the absolute value,  F is equal to  1 on all the integers, and ostthlem1 20829 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
  n ) )   &    |-  ( ph  ->  A. n  e. 
 Prime  -.  ( F `  n )  <  1 )   =>    |-  ( ph  ->  F  =  K )
 
Theoremostth2lem2 20836* Lemma for ostth2 20839. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   =>    |-  ( ( ph  /\  X  e.  NN0  /\  Y  e.  ( 0 ... (
 ( M ^ X )  -  1 ) ) )  ->  ( F `  Y )  <_  (
 ( M  x.  X )  x.  ( T ^ X ) ) )
 
Theoremostth2lem3 20837* Lemma for ostth2 20839. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   &    |-  U  =  ( ( log `  N )  /  ( log `  M )
 )   =>    |-  ( ( ph  /\  X  e.  NN )  ->  (
 ( ( F `  N )  /  ( T  ^ c  U ) ) ^ X ) 
 <_  ( X  x.  (
 ( M  x.  T )  x.  ( U  +  1 ) ) ) )
 
Theoremostth2lem4 20838* Lemma for ostth2 20839. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   &    |-  U  =  ( ( log `  N )  /  ( log `  M )
 )   =>    |-  ( ph  ->  (
 1  <  ( F `  M )  /\  R  <_  S ) )
 
Theoremostth2 20839* - Lemma for ostth 20841: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   =>    |-  ( ph  ->  E. a  e.  ( 0 (,] 1
 ) F  =  ( y  e.  QQ  |->  ( ( abs `  y
 )  ^ c  a ) ) )
 
Theoremostth3 20840* - Lemma for ostth 20841: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
  n ) )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( F `  P )  <  1 )   &    |-  R  =  -u ( ( log `  ( F `  P ) )  /  ( log `  P ) )   &    |-  S  =  if (
 ( F `  P )  <_  ( F `  p ) ,  ( F `  p ) ,  ( F `  P ) )   =>    |-  ( ph  ->  E. a  e.  RR+  F  =  ( y  e.  QQ  |->  ( ( ( J `  P ) `  y
 )  ^ c  a ) ) )
 
Theoremostth 20841* Ostrowski's theorem, which classifies all absolute values on  QQ. Any such absolute value must either be the trivial absolute value  K, a constant exponent  0  <  a  <_  1 times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   =>    |-  ( F  e.  A  <->  ( F  =  K  \/  E. a  e.  ( 0 (,] 1 ) F  =  ( y  e. 
 QQ  |->  ( ( abs `  y )  ^ c  a ) )  \/ 
 E. a  e.  RR+  E. g  e.  ran  J  F  =  ( y  e.  QQ  |->  ( ( g `
  y )  ^ c  a ) ) ) )
 
PART 14  GUIDES AND MISCELLANEA
 
14.1  Guides (conventions, explanations, and examples)
 
14.1.1  Conventions

This section describes the conventions we use. However, these conventions often refer to existing mathematical practices, which are discussed in more detail in other references. Logic and set theory provide a foundation for all of mathematics. To learn about them, you should study one or more of the references listed below. We indicate references using square brackets. The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

  • Axioms of propositional calculus - [Margaris].
  • Axioms of predicate calculus - [Megill] (System S3' in the article referenced).
  • Theorems of propositional calculus - [WhiteheadRussell].
  • Theorems of pure predicate calculus - [Margaris].
  • Theorems of equality and substitution - [Monk2], [Tarski], [Megill].
  • Axioms of set theory - [BellMachover].
  • Development of set theory - [TakeutiZaring]. (The first part of [Quine] has a good explanation of the powerful device of "virtual" or class abstractions, which is essential to our development.)
  • Construction of real and complex numbers - [Gleason]
  • Theorems about real numbers - [Apostol]
 
Theoremconventions 20842 Here are some of the conventions we use in the Metamath Proof Explorer (aka "set.mm"), and how they correspond to typical textbook language (skipping the many cases where they're identical):

  • Notation. Where possible, the notation attempts to conform to modern conventions, with variations due to our choice of the axiom system or to make proofs shorter. However, our notation is strictly sequential (left-to-right). For example, summation is written in the form  sum_ k  e.  A B (df-sum 12206) which denotes that index variable  k ranges over  A when evaluating  B. Thus,  sum_ k  e.  NN  ( 1  /  ( 2 ^ k ) )  =  1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 12385). Also, the method of definition, the axioms for predicate calculus, and the development of substitution are somewhat different from those found in standard texts. For example, the expressions for the axioms were designed for direct derivation of standard results without excessive use of metatheorems. (See Theorem 9.7 of [Megill] p. 448 for a rigorous justification.) The notation is usually explained in more detail when first introduced.
  • Axiomatic assertions ($a). All axiomatic assertions ($a statements) starting with "  |-" have labels starting with "ax-" (axioms) or "df-" (definitions). A statement with a label starting with "ax-" corresponds to what is traditionally called an axiom. A statement with a label starting with "df-" introduces new symbols or a new relationship among symbols that can be eliminated; they always extend the definition of a wff or class. Metamath blindly treats $a statements as new given facts but does not try to justify them. The mmj2 program will justify the definitions as sound, except for 4 (df-bi, df-cleq, df-clel, df-clab) that require a more complex metalogical justification by hand.
  • Proven axioms. In some cases we wish to treat an expression as an axiom in later theorems, even though it can be proved. For example, we derive the postulates or axioms of complex arithmetic as theorems of ZFC set theory. For convenience, after deriving the postulates we re-introduce them as new axioms on top of set theory. This lets us easily identify which axioms are needed for a particular complex number proof, without the obfuscation of the set theory used to derive them. For more, see http://us.metamath.org/mpeuni/mmcomplex.html. When we wish to use a previously-proven assertion as an axiom, our convention is that we use the regular "ax-NAME" label naming convention to define the axiom, but we precede it with a proof of the same statement with the label "axNAME" . An example is complex arithmetic axiom ax-1cn 8840, proven by the preceding theorem ax1cn 8816. The metamath.exe program will warn if an axiom does not match the preceding theorem that justifies it if the names match in this way.
  • Definitions (df-...). We encourage definitions to include hypertext links to proven examples.
  • Statements with hypotheses. Many theorems and some axioms, such as ax-mp 8, have hypotheses that must be satisfied in order for the conclusion to hold, in this case min and maj. When presented in summarized form such as in the Theorem List (click on "Nearby theorems" on the ax-mp 8 page), the hypotheses are connected with an ampersand and separated from the conclusion with a big arrow, such as in "  |-  ph &  |-  ( ph  ->  ps ) =>  |-  ps". These symbols are not part of the Metamath language but are just informal notation meaning "and" and "implies".
  • Discouraged use and modification. If something should only be used in limited ways, it is marked with "(New usage is discouraged.)". This is used, for example, when something can be constructed in more than one way, and we do not want later theorems to depend on that specific construction. This marking is also used if we want later proofs to use proven axioms. For example, we want later proofs to use ax-1cn 8840 (not ax1cn 8816) and ax-1ne0 8851 (not ax1ne0 8827), as these are proven axioms for complex arithmetic. Thus, both ax1cn 8816 and ax1ne0 8827 are marked as "(New usage is discouraged.)". In some cases a proof should not normally be changed, e.g., when it demonstrates some specific technique. These are marked with "(Proof modification is discouraged.)".
  • New definitions infrequent. Typically, we are minimalist when introducing new definitions; they are introduced only when a clear advantage becomes apparent for reducing the number of symbols, shortening proofs, etc. We generally avoid the introduction of gratuitous definitions because each one requires associated theorems and additional elimination steps in proofs. For example, we use  < and  <_ for inequality expressions, and use  ( ( sin `  ( _i  x.  A ) )  /  _i ) instead of  (sinh `  A ) for the hyperbolic sine.
  • Axiom of choice. The axiom of choice (df-ac 7788) is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. Therefore, where a proof does not require the axiom of choice, we prefer that proof instead. E.g., our proof of the Schroeder-Bernstein Theorem (sbth 7024) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 8106) or axiom of dependent choice (ax-dc 8117) can be used instead.
  • Variables. Typically, Greek letters ( ph = phi,  ps = psi,  ch = chi, etc.),... are used for propositional (wff) variables;  x,  y,  z,... for individual (set) variables; and  A,  B,  C,... for class variables.
  • Turnstile. " |-", meaning "It is provable that," is the first token of all assertions and hypotheses that aren't syntax constructions. This is a standard convention in logic. For us, it also prevents any ambiguity with statements that are syntax constructions, such as "wff  -.  ph".
  • Biconditional ( <->). There are basically two ways to maximize the effectiveness of biconditionals ( <->): you can either have one-directional simplifications of all theorems that produce biconditionals, or you can have one-directional simplifications of theorems that consume biconditionals. Some tools (like Lean) follow the first approach, but set.mm follows the second approach. Practically, this means that in set.mm, for every theorem that uses an implication in the hypothesis, like ax-mp 8, there is a corresponding version with a biconditional or a reversed biconditional, like mpbi 199 or mpbir 200. We prefer this second approach because the number of duplications in the second approach is bounded by the size of the propositional calculus section, which is much smaller than the number of possible theorems in all later sections that produce biconditionals. So although theorems like biimpi 186 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 886, sylbir 204, or 3imtr4i 257.
  • Substitution. " [ y  /  x ] ph" should be read "the wff that results from the proper substitution of  y for  x in wff  ph." See df-sb 1640 and the related df-sbc 3026 and df-csb 3116.
  • Is-a set. " A  e.  _V" should be read "Class  A is a set (i.e. exists)." This is a convenient convention based on Definition 2.9 of [Quine] p. 19. See df-v 2824 and isset 2826.
  • Converse. " `' R" should be read "converse of (relation)  R" and is the same as the more standard notation R^{-1} (the standard notation is ambiguous). See df-cnv 4734. This can be used to define a subset, e.g., df-tan 12400 notates "the set of values whose cosine is a nonzero complex number" as  ( `' cos " ( CC  \  { 0 } ) ).
  • Function application. "( F `  x)" should be read "the value of function  F at  x" and has the same meaning as the more familiar but ambiguous notation F(x). For example,  ( cos `  0 )  =  1 (see cos0 12477). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. See df-fv 5300. In the ASCII (input) representation there are spaces around the grave accent; there is a single accent when it is used directly, and it is doubled within comments.
  • Infix and parentheses. When a function that takes two classes and produces a class is applied as part of an infix expression, the expression is always surrounded by parentheses (see df-ov 5903). For example, the  + in  ( 2  +  2 ); see 2p2e4 9889. Function application is itself an example of this. Similarly, predicate expressions in infix form that take two or three wffs and produce a wff are also always surrounded by parentheses, such as  ( ph  ->  ps ),  ( ph  \/  ps ),  ( ph  /\  ps ), and  ( ph  <->  ps ) (see wi 4, df-or 359, df-an 360, and df-bi 177 respectively). In contrast, a binary relation (which compares two classes and produces a wff) applied in an infix expression is not surrounded by parentheses. This includes set membership  A  e.  B (see wel 1702), equality  A  =  B (see df-cleq 2309), subset  A  C_  B (see df-ss 3200), and less-than  A  <  B (see df-lt 8795). For the general definition of a binary relation in the form  A R B, see df-br 4061. For example,  0  <  1 ( see 0lt1 9341) does not use parentheses.
  • Unary minus. The symbol  -u is used to indicate a unary minus, e.g.,  -u 1. It is specially defined because it is so commonly used. See cneg 9083.
  • Function definition. Functions are typically defined by first defining the constant symbol (using $c) and declaring that its symbol is a class with the label cNAME (e.g., ccos 12393). The function is then defined labelled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 12399). Typically, there are other proofs such as its closure labelled NAMEcl (e.g., coscl 12454), its function application form labelled NAMEval (e.g., cosval 12450), and at least one simple value (e.g., cos0 12477).
  • Factorial. The factorial function is traditionally a postfix operation, but we treat it as a normal function applied in prefix form, e.g.,  ( ! `  4 )  = ; 2 4 (df-fac 11336 and fac4 11343).
  • Unambiguous symbols. A given symbol has a single unambiguous meaning in general. Thus, where the literature might use the same symbol with different meanings, here we use different (variant) symbols for different meanings. These variant symbols often have suffixes, subscripts, or underlines to distinguish them. For example, here " 0" always means the value zero (df-0 8789), while " 0g" is the group identity element (df-0g 13453), " 0." is the poset zero (df-p0 14194), " 0 p" is the zero polynomial (df-0p 19078), " 0vec" is the zero vector in a normed complex vector space (df-0v 21209), and " .0." is a class variable for use as a connective symbol (this is used, for example, in p0val 14196). There are other class variables used as connective symbols where traditional notation would use ambiguous symbols, including " .1.", " .+", " .*", and " .||". These symbols are very similar to traditional notation, but because they are different symbols they eliminate ambiguity.
  • Natural numbers. There are different definitions of "natural" numbers in the literature. We use  NN (df-nn 9792) for the integer numbers starting from 1, and  NN0 (df-n0 10013) for the set of nonnegative integers starting at zero.
  • Decimal numbers. Numbers larger than ten are often expressed in base 10 using the decimal constructor df-dec 10172, e.g., ;;; 4 0 0 1 (see 4001prm 13190 for a proof that 4001 is prime).
  • Theorem forms. We often call a theorem a "deduction" whenever the conclusion and all hypotheses are each prefixed with the same antecedent  ph  ->. Deductions are often the preferred form for theorems because they allow us to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem would be used. See, for example, a1d 22. A deduction hypothesis can have an indirect antecedent via definitions, e.g., see lhop 19416. Deductions have a label suffix of "d" if there are other forms of the same theorem. By contrast, we tend to call the simpler version with no common antecedent an "inference" and suffix its label with "i"; compare theorem a1i 10. Finally, a "tautology" would be the form with no hypotheses, and its label would have no suffix. For example, see pm2.43 47, pm2.43i 43, and pm2.43d 44.
  • Deduction theorem. The Deduction Theorem is a metalogical theorem that provides an algorithm for constructing a proof of a theorem from the proof of its corresponding deduction. In ordinary mathematics, no one actually carries out the algorithm, because (in its most basic form) it involves an exponential explosion of the number of proof steps as more hypotheses are eliminated. Instead, in ordinary mathematics the Deduction Theorem is invoked simply to claim that something can be done in principle, without actually doing it. For more details, see http://us.metamath.org/mpeuni/mmdeduction.html. The Deduction Theorem is a metalogical theorem that cannot be applied directly in metamath, and the explosion of steps would be a problem anyway, so alternatives are used. One alternative we use sometimes is the "weak deduction theorem" dedth 3640, which works in certain cases in set theory. We also sometimes use dedhb 2969. However, the primary mechanism we use today for emulating the deduction theorem is to write proofs in the deduction theorem form (aka "deduction style") described earlier; the prefixed  ph  -> mimics the context in a deduction proof system. In practice this mechanism works very well. This approach is described in the deduction form and natural deduction page; a list of translations for common natural deduction rules is given in natded 20843.
  • Recursion. We define recursive functions using various "recursion constructors". These allow us to define, with compact direct definitions, functions that are usually defined in textbooks with indirect self-referencing recursive definitions. This produces compact definition and much simpler proofs, and greatly reduces the risk of creating unsound definitions. Examples of recursion constructors include recs ( F ) in df-recs 6430,  rec ( F ,  I ) in df-rdg 6465, seq𝜔 ( F ,  I ) in df-seqom 6502, and  seq  M (  .+  ,  F ) in df-seq 11094. These have characteristic function  F and initial value  I. ( gsumg in df-gsum 13454 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 6430, but for the "average user" the most useful one is probably df-seq 11094- provided that a countable sequence is sufficient for the recursion.
  • Extensible structures. Mathematics includes many structures such as ring, group, poset, etc. We define an "extensible structure" which is then used to define group, ring, poset, etc. This allows theorems from more general structures (groups) to be reused for more specialized structures (rings) without having to reprove them. See df-struct 13197.
  • Organizing proofs. Humans have trouble understanding long proofs. It is often preferable to break longer proofs into smaller parts (just as with traditional proofs). In Metamath this is done by creating separate proofs of the separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof's name followed by "lem", and a number if there is more than one. E.g., sbthlem1 7014 is the first lemma for sbth 7024. Also, consider proving reusable results separately, so that others will be able to easily reuse that part of your work.
  • Hypertext links. We strongly encourage comments to have many links to related material, with accompanying text that explains the relationship. These can help readers understand the context. Links to other statements, or to HTTP/HTTPS URLs, can be inserted in ASCII source text by prepending a space-separated tilde. When metamath.exe is used to generate HTML it automatically inserts hypertext links for syntax used (e.g., every symbol used), every axiom and definition depended on, the justification for each step in a proof, and to both the next and previous assertion.
  • Bibliography references. Please include a bibliographic reference to any external material used. A name in square brackets in a comment indicates a bibliographic reference. The full reference must be of the form KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER - note that this is a very specific form that requires a page number. There should be no comma between the author reference and the "p." (a constant indicator). Whitespace, comma, period, or semicolon should follow NUMBER. An example is Theorem 3.1 of [Monk1] p. 22, The KEYWORD, which is not case-sensitive, must be one of the following: Axiom, Chapter, Compare, Condition, Corollary, Definition, Equation, Example, Exercise, Figure, Item, Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem, Property, Proposition, Remark, Rule, Scheme, Section, or Theorem. The IDENTIFIER is optional, as in for example "Remark in [Monk1] p. 22". The NOISEWORDS(S) are zero or more from the list: from, in, of, on. The AUTHOR(S) must be present in the file identified with the htmlbibliography assignment (e.g. mmset.html) as a named anchor (NAME=). If there is more than one document by the same author(s), add a numeric suffix (as shown here). The NUMBER is a page number, and may be any alphanumeric string such as an integer or Roman numeral. Note that we require page numbers in comments for individual $a or $p statements. We allow names in square brackets without page numbers (a reference to an entire document) in heading comments. If this is a new reference, please also add it to the "Bibliography" section of mmset.html. (The file mmbiblio.html is automatically rebuilt, e.g., using the metamath.exe "write bibliography" command.)
  • Input format. The input is in ASCII with two-space indents. Tab characters are not allowed. Use embedded math comments or HTML entities for non-ASCII characters (e.g., "&eacute;" for "é").
  • Information on syntax, axioms, and definitions. For a hyperlinked list of syntax, axioms, and definitions, see http://us.metamath.org/mpeuni/mmdefinitions.html. If you have questions about a specific symbol or axiom, it is best to go directly to its definition to learn more about it. The generated HTML for each theorem and axiom includes hypertext links to each symbol's definition.

Naming conventions

Every statement has a unique identifying label. We use various label naming conventions to provide easy-to-remember hints about their contents. Labels are not a 1-to-1 mapping, because that would create long names that would be difficult to remember and tedious to type. Instead, label names are relatively short while suggesting their purpose. Names are occasionally changed to make them more consistent or as we find better ways to name them. Here are a few of the label naming conventions:

  • Axioms, definitions, and wff syntax. As noted earlier, axioms are named "ax-NAME", proofs of proven axioms are named "axNAME", and definitions are named "df-NAME". Wff syntax declarations have labels beginning with "w" followed by short fragment suggesting its purpose.
  • Hypotheses. Hypotheses have the name of the final axiom or theorem, followed by ".", followed by a unique id.
  • Common names. If a theorem has a well-known name, that name (or a short version of it) is sometimes used directly. Examples include barbara 2273 and stirling 26986.
  • Syntax label fragments. Most theorems are named using syntax label fragments. Almost every syntactic construct has a definition labelled "df-NAME", and NAME normally is the syntax label fragment. For example, the difference construct  ( A  \  B ) is defined in df-dif 3189, and thus its syntax label fragment is "dif". Similarly, the singleton construct  { A } has syntax label fragment "sn" (because it is defined in df-sn 3680), the subclass (subset) relation  A  C_  B has "ss" (because it is defined in df-ss 3200), and the pair construct  { A ,  B } has "pr" (df-pr 3681). Theorem names are often a concatenation of the syntax label fragments (omitting variables). For example, a theorem about  ( A  \  B )  C_  A involves a difference ("dif") of a subset ("ss"), and thus is named difss 3337. Digits are used to represent themselves. Suffixes (e.g., with numbers) are sometimes used to distinguish multiple theorems that would otherwise produce the same label.
  • Phantom definitions. In some cases there are common label fragments for something that could be in a definition, but for technical reasons is not. The is-element-of (is member of) construct  A  e.  B does not have a df-NAME definition; in this case its syntax label fragment is "el". Thus, because the theorem beginning with  ( A  e.  ( B  \  { C } ) uses is-element-of ("el") of a difference ("dif") of a singleton ("sn"), it is named eldifsn 3783. An "n" is often used for negation ( -.), e.g., nan 563.
  • Exceptions. Sometimes there is a definition df-NAME but the label fragment is not the NAME part. The table below attempts to list all such cases and marks them in bold. For example, label fragment "cn" represents complex numbers  CC (even though its definition is in df-c 8788) and "re" represents real numbers  RR. The empty set  (/) often uses fragment 0, even though it is defined in df-nul 3490. Syntax construct  ( A  +  B ) usually uses the fragment "add" (which is consistent with df-add 8793), but "p" is used as the fragment for constant theorems. Equality  ( A  =  B ) often uses "e" as the fragment. As a result, "two plus two equals four" is named 2p2e4 9889.
  • Other markings. In labels we sometimes use "com" for "commutative", "ass" for "associative", "rot" for "rotation", and "di" for "distributive".
  • Principia Mathematica. Proofs of theorems from Principia Mathematica often use a different naming convention. They are instead often named "pm" followed by its identifier. For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named pm2.27 35.
  • Closures and values. As noted above, if a function df-NAME is defined, there is typically a proof of its value named "NAMEval" and its closure named "NAMEcl". E.g., for cosine (df-cos 12399) we have value cosval 12450 and closure coscl 12454.
  • Special cases. Sometimes syntax and related markings are insufficient to distinguish different theorems. For example, there are over 100 different implication-exclusive theorems. These are then grouped in a more ad-hoc way that attempts to make their distinctions clearer. These often use abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and "id" for "identity". It's especially hard to give good names in the propositional calculus section because there are so few primitives. However, in most cases this is not a serious problem. There are a few very common theorems like ax-mp 8 and syl 15 that you will have no trouble remembering, a few theorem series like syl*anc and simp* that you can use parametrically, and a few other useful glue things for destructuring 'and's and 'or's (see natded 20843 for a list), and that's about all you need for most things. As for the rest, you can just assume that if it involves three or less connectives we probably already have a proof, and searching for it will give you the name.
  • Suffixes. We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as  F/ x ph in 19.21 1822 via the use of distinct variable conditions combined with nfv 1610. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 2182 derived from df-eu 2180. The "f" stands for "not free in" which is less restrictive than "does not occur in." We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 15) -type inference in a proof. When an inference is converted to a theorem by eliminating an "is a set" hypothesis, we sometimes suffix the theorem form with "g" (for "more general") as in uniex 4553 vs. uniexg 4554. A theorem name is suffixed with "ALT" if it's an alternative less-preferred proof of a theorem.
  • Reuse. When creating a new theorem or axiom, try to reuse abbreviations used elsewhere. A comment should explain the first use of an abbreviation.

The following table shows some commonly-used abbreviations in labels, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME. This is not a complete list of abbreviations, though we do want this to eventually be a complete list of exceptions.

AbbreviationMnenomicSource ExpressionSyntax?Example(s)
addadd (see "p") df-add 8793  ( A  +  B ) Yes addcl 8864, addcom 9043, addass 8869
ALTalternative/less preferred (suffix) No
anand df-an 360  ( ph  /\  ps ) Yes anor 475, iman 413, imnan 411
assassociative No biass 348, orass 510, mulass 8870
bibiconditional df-bi 177  ( ph  <->  ps ) Yes impbid 183
cncomplex numbers df-c 8788  CC Yes nnsscn 9796, nncn 9799
comcommutative No orcom 376, bicomi 193, eqcomi 2320
ddeduction form No idd 21, impbid 183
di, distrdistributive No andi 837, imdi 352, ordi 834, difindi 3457, ndmovdistr 6051
difdifference df-dif 3189  ( A  \  B ) Yes difss 3337, difindi 3457
divdivision df-div 9469  ( A  /  B ) Yes divcl 9475, divval 9471, divmul 9472
e, eqequals df-cleq 2309  A  =  B Yes 2p2e4 9889, uneqri 3351
elelement of  A  e.  B Yes eldif 3196, eldifsn 3783, elssuni 3892
f"not free in" (suffix) No
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 4554
ididentity No
idmidempotent No anidm 625, tpidm13 3763
im, impimplication (label often omitted) df-im 11633  ( A  ->  B ) Yes iman 413, imnan 411, impbidd 181
inintersection df-in 3193  ( A  i^i  B ) Yes elin 3392, incom 3395
is...is (something a) ...? No isrng 15394
mpmodus ponens ax-mp 8 No mpd 14, mpi 16
mulmultiplication (see "t") df-mul 8794  ( A  x.  B ) Yes mulcl 8866, divmul 9472, mulcom 8868, mulass 8870
n, notnot  -.  ph Yes nan 563, notnot2 104
ne0not equal to zero (see n0)  =/=  0 No negne0d 9200, ine0 9260, gt0ne0 9284
nnnatural numbers df-nn 9792  NN Yes nnsscn 9796, nncn 9799
n0not the empty set (see ne0)  =/=  (/) No n0i 3494, vn0 3496, ssn0 3521
oror df-or 359  ( ph  \/  ps ) Yes orcom 376, anor 475
pplus (see "add"), for all-constant theorems df-add 8793  ( 3  +  2 )  =  5 Yes 3p2e5 9902
pmPrincipia Mathematica No pm2.27 35
prpair df-pr 3681  { A ,  B } Yes elpr 3692, prcom 3739, prid1g 3766, prnz 3779
q  QQ (quotients) df-q 10364  QQ Yes elq 10365
rereal numbers df-r 8792  RR Yes recn 8872, 0re 8883
rngring df-rng 15389  Ring Yes rngidval 15392, isrng 15394, rnggrp 15395
rotrotation No 3anrot 939, 3orrot 940
seliminates need for syllogism (suffix) No
snsingleton df-sn 3680  { A } Yes eldifsn 3783
sssubset df-ss 3200  A  C_  B Yes difss 3337
subsubtract df-sub 9084  ( A  -  B ) Yes subval 9088, subaddi 9178
sylsyllogism syl 15 No 3syl 18
t times (see "mul"), for all-constant theorems df-mul 8794  ( 3  x.  2 )  =  6 Yes 3t2e6 9919
tptriple df-tp 3682  { A ,  B ,  C } Yes eltpi 3711, tpeq1 3749
ununion df-un 3191  ( A  u.  B ) Yes uneqri 3351, uncom 3353
vdistinct variable conditions used when a not-free hypothesis (suffix) No spimv 1962
xreXtended reals df-xr 8916  RR* Yes ressxr 8921, rexr 8922, 0xr 8923
z  ZZ (integers, from German Zahlen) df-z 10072  ZZ Yes elz 10073, zcn 10076
0, z slashed zero (empty set) (see n0) df-nul 3490  (/) Yes n0i 3494, vn0 3496; snnz 3778, prnz 3779

Distinctness or freeness

Here are some conventions that address distinctness or freeness of a variable:

  •  F/ x ph is read "  x is not free in (wff)  ph"; see df-nf 1536 (whose description has some important technical details). Similarly,  F/_ x A is read  x is not free in (class)  A, see df-nfc 2441.
  • "$d x y $." should be read "Assume x and y are distinct variables."
  • "$d x  ph $." should be read "Assume x does not occur in phi $." Sometimes a theorem is proved using  F/ x ph (df-nf 1536) in place of "$d  x ph $." when a more general result is desired; ax-17 1607 can be used to derive the $d version. For an example of how to get from the $d version back to the $e version, see the proof of euf 2182 from df-eu 2180.
  • "$d x A $." should be read "Assume x is not a variable occurring in class A."
  • "$d x A $. $d x ps $. $e |-  ( x  =  A  ->  ( ph  <->  ps ) ) $." is an idiom often used instead of explicit substitution, meaning "Assume psi results from the proper substitution of A for x in phi."
  • "  |-  ( -.  A. x x  =  y  ->  ..." occurs early in some cases, and should be read "If x and y are distinct variables, then..." This antecedent provides us with a technical device (called a "distinctor" in Section 7 of [Megill] p. 444) to avoid the need for the $d statement early in our development of predicate calculus, permitting unrestricted substitutions as conceptually simple as those in propositional calculus. However, the $d eventually becomes a requirement, and after that this device is rarely used.

Here is more information about our processes for checking and contributing to this work:

  • Multiple verifiers. This entire file is verified by multiple independently-implemented verifiers when it is checked in, giving us extremely high confidence that all proofs follow from the assumptions. The checkers also check for various other problems such as overly long lines.
  • Rewrapped line length. The input file routinely has its text wrapped using metamath 'read set.mm' 'save proof */c/f' 'write source set.mm/rewrap' (so please do the same).
  • Discouraged information. A separate file named "discouraged" lists all discouraged statements and uses of them, and this file is checked. If you change the use of discouraged things, you will need to change this file. This makes it obvious when there is a change to anything discouraged (triggering further review).
  • Proposing specific changes. Please propose specific changes as pull requests (PRs) against the "develop" branch of set.mm, at: https://github.com/metamath/set.mm/tree/develop
  • Community. We encourage anyone interested in Metamath to join our mailing list: https://groups.google.com/forum/#!forum/metamath.

(Contributed by DAW, 27-Dec-2016.)

 |-  ph   =>    |-  ph
 
14.1.2  Natural deduction
 
Theoremnatded 20843 Here are typical natural deduction (ND) rules in the style of Gentzen and Jaśkowski, along with MPE translations of them. This also shows the recommended theorems when you find yourself needing these rules (the recommendations encourage a slightly different proof style that works more naturally with metamath). A decent list of the standard rules of natural deduction can be found beginning with definition /\I in [Pfenning] p. 18. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. Many more citations could be added.

NameNatural Deduction RuleTranslation RecommendationComments
IT  _G |-  ps =>  _G |-  ps idi 2 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
 /\I  _G |-  ps &  _G |-  ch =>  _G |-  ps  /\  ch jca 518 jca 518, pm3.2i 441 Definition  /\I in [Pfenning] p. 18, definition I /\m,n in [Clemente] p. 10, and definition  /\I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 /\EL  _G |-  ps  /\  ch =>  _G |-  ps simpld 445 simpld 445, adantr 451 Definition  /\EL in [Pfenning] p. 18, definition E /\(1) in [Clemente] p. 11, and definition  /\E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 /\ER  _G |-  ps  /\  ch =>  _G |-  ch simprd 449 simpr 447, adantl 452 Definition  /\ER in [Pfenning] p. 18, definition E /\(2) in [Clemente] p. 11, and definition  /\E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 ->I  _G ,  ps |-  ch =>  _G |-  ps  ->  ch ex 423 ex 423 Definition  ->I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition  ->I in [Indrzejczak] p. 33.
 ->E  _G |-  ps  ->  ch &  _G |-  ps =>  _G |-  ch mpd 14 ax-mp 8, mpd 14, mpdan 649, imp 418 Definition  ->E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition  ->E in [Indrzejczak] p. 33.
 \/IL  _G |-  ps =>  _G |-  ps  \/  ch olcd 382 olc 373, olci 380, olcd 382 Definition  \/I in [Pfenning] p. 18, definition I \/n(1) in [Clemente] p. 12
 \/IR  _G |-  ch =>  _G |-  ps  \/  ch orcd 381 orc 374, orci 379, orcd 381 Definition  \/IR in [Pfenning] p. 18, definition I \/n(2) in [Clemente] p. 12.
 \/E  _G |-  ps  \/  ch &  _G ,  ps |-  th &  _G ,  ch |-  th =>  _G |-  th mpjaodan 761 mpjaodan 761, jaodan 760, jaod 369 Definition  \/E in [Pfenning] p. 18, definition E \/m,n,p in [Clemente] p. 12.
 -.I  _G ,  ps |-  F. =>  _G |-  -.  ps inegd 1324 pm2.01d 161
 -.I  _G ,  ps |-  th &  _G |-  -.  th =>  _G |-  -.  ps mtand 640 mtand 640 definition I -.m,n,p in [Clemente] p. 13.
 -.I  _G ,  ps |-  ch &  _G ,  ps |-  -.  ch =>  _G |-  -.  ps pm2.65da 559 pm2.65da 559 Contradiction.
 -.I  _G ,  ps |-  -.  ps =>  _G |-  -.  ps pm2.01da 429 pm2.01d 161, pm2.65da 559, pm2.65d 166 For an alternative falsum-free natural deduction ruleset
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |-  F. pm2.21fal 1326 pm2.21dd 99
 -.E  _G ,  -.  ps |-  F. =>  _G |-  ps pm2.21dd 99 definition  ->E in [Indrzejczak] p. 33.
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |-  th pm2.21dd 99 pm2.21dd 99, pm2.21d 98, pm2.21 100 For an alternative falsum-free natural deduction ruleset. Definition  -.E in [Pfenning] p. 18.
 T.I  _G |-  T. a1tru 1321 tru 1312, a1tru 1321, trud 1314 Definition  T.I in [Pfenning] p. 18.
 F.E  _G ,  F.  |-  th falimd 1320 falim 1319 Definition  F.E in [Pfenning] p. 18.
 A.I  _G |-  [ a  /  x ] ps =>  _G |-  A. x ps alrimiv 1622 alrimiv 1622, ralrimiva 2660 Definition  A.Ia in [Pfenning] p. 18, definition I A.n in [Clemente] p. 32.
 A.E  _G |-  A. x ps =>  _G |-  [ t  /  x ] ps spsbcd 3038 spcv 2908, rspcv 2914 Definition  A.E in [Pfenning] p. 18, definition E A.n,t in [Clemente] p. 32.
 E.I  _G |-  [ t  /  x ] ps =>  _G |-  E. x ps spesbcd 3107 spcev 2909, rspcev 2918 Definition  E.I in [Pfenning] p. 18, definition I E.n,t in [Clemente] p. 32.
 E.E  _G |-  E. x ps &  _G ,  [ a  /  x ] ps |-  th =>  _G |-  th exlimddv 1629 exlimddv 1629, exlimdd 1861, exlimdv 1627, rexlimdva 2701 Definition  E.Ea,u in [Pfenning] p. 18, definition E E.m,n,p,a in [Clemente] p. 32.
 F.C  _G ,  -.  ps |-  F. =>  _G |-  ps efald 1325 efald 1325 Proof by contradiction (classical logic), definition  F.C in [Pfenning] p. 17.
 F.C  _G ,  -.  ps |-  ps =>  _G |-  ps pm2.18da 430 pm2.18da 430, pm2.18d 103, pm2.18 102 For an alternative falsum-free natural deduction ruleset
 -.  -.C  _G |-  -.  -.  ps =>  _G |-  ps notnotrd 105 notnotrd 105, notnot2 104 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E -.n in [Clemente] p. 14.
EM  _G |-  ps  \/  -.  ps exmidd 405 exmid 404 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
 =I  _G |-  A  =  A eqidd 2317 eqid 2316, eqidd 2317 Introduce equality, definition =I in [Pfenning] p. 127.
 =E  _G |-  A  =  B &  _G [. A  /  x ]. ps =>  _G |-  [. B  /  x ]. ps sbceq1dd 3031 sbceq1d 3030, equality theorems Eliminate equality, definition =E in [Pfenning] p. 127. (Both E1 and E2.)

Note that MPE uses classical logic, not intuitionist logic. As is conventional, the "I" rules are introduction rules, "E" rules are elimination rules, the "C" rules are conversion rules, and  _G represents the set of (current) hypotheses. We use wff variable names beginning with  ps to provide a closer representation of the Metamath equivalents (which typically use the antedent  ph to represent the context  _G).

Most of this information was developed by Mario Carneiro and posted on 3-Feb-2017. For more information, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer.

For annotated examples where some traditional ND rules are directly applied in MPE, see ex-natded5.2 20844, ex-natded5.3 20847, ex-natded5.5 20850, ex-natded5.7 20851, ex-natded5.8 20853, ex-natded5.13 20855, ex-natded9.20 20857, and ex-natded9.26 20859.

(Contributed by DAW, 4-Feb-2017.)

 |-  ph   =>    |-  ph
 
14.1.3  Natural deduction examples

These are examples of how natural deduction rules can be applied in metamath (both as line-for-line translations of ND rules, and as a way to apply deduction forms without being limited to applying ND rules). For more information, see natded 20843 and http://us.metamath.org/mpeuni/mmnatded.html.

 
Theoremex-natded5.2 20844 Theorem 5.2 of [Clemente] p. 15, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows:
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
15  ( ( ps  /\  ch )  ->  th )  ( ph  ->  ( ( ps  /\  ch )  ->  th ) ) Given $e.
22  ( ch  ->  ps )  ( ph  ->  ( ch  ->  ps ) ) Given $e.
31  ch  ( ph  ->  ch ) Given $e.
43  ps  ( ph  ->  ps )  ->E 2,3 mpd 14, the MPE equivalent of  ->E, 1,2
54  ( ps  /\  ch )  ( ph  ->  ( ps  /\  ch ) )  /\I 4,3 jca 518, the MPE equivalent of  /\I, 3,1
66  th  ( ph  ->  th )  ->E 1,5 mpd 14, the MPE equivalent of  ->E, 4,5

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps). A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.2-2 20845. A proof without context is shown in ex-natded5.2i 20846. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2-2 20845 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 20844 and ex-natded5.2i 20846. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2i 20846 The same as ex-natded5.2 20844 and ex-natded5.2-2 20845 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ps  /\  ch )  ->  th )   &    |-  ( ch  ->  ps )   &    |-  ch   =>    |- 
 th
 
Theoremex-natded5.3 20847 Theorem 5.3 of [Clemente] p. 16, translated line by line using an interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.3-2 20848. A proof without context is shown in ex-natded5.3i 20849. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 451 to move it into the ND hypothesis
25;6  ( ch  ->  th )  ( ph  ->  ( ch  ->  th ) ) Given $e; adantr 451 to move it into the ND hypothesis
31 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 447, to access the new assumption
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 14, the MPE equivalent of  ->E, 1.3. adantr 451 was used to transform its dependency (we could also use imp 418 to get this directly from 1)
57 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 14, the MPE equivalent of  ->E, 4,6. adantr 451 was used to transform its dependency
68 ...  ( ch  /\  th )  ( ( ph  /\  ps )  ->  ( ch  /\  th ) )  /\I 4,5 jca 518, the MPE equivalent of  /\I, 4,7
79  ( ps  ->  ( ch  /\  th ) )  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )  ->I 3,6 ex 423, the MPE equivalent of  ->I, 8

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3-2 20848 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 20847 and ex-natded5.3i 20849. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3i 20849 The same as ex-natded5.3 20847 and ex-natded5.3-2 20848 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ps  ->  ch )   &    |-  ( ch  ->  th )   =>    |-  ( ps  ->  ( ch  /\  th ) )
 
Theoremex-natded5.5 20850 Theorem 5.5 of [Clemente] p. 18, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 451 to move it into the ND hypothesis
25  -.  ch  ( ph  ->  -.  ch ) Given $e; we'll use adantr 451 to move it into the ND hypothesis
31 ...|  ps  ( ph  ->  ps ) ND hypothesis assumption simpr 447
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 14 1,3
56 ...  -.  ch  ( ( ph  /\  ps )  ->  -.  ch ) IT 2 adantr 451 5
67  -.  ps  ( ph  ->  -.  ps )  /\I 3,4,5 pm2.65da 559 4,6

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 451; simpr 447 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is mtod 168; a proof without context is shown in mto 167.

(Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.7 20851 Theorem 5.7 of [Clemente] p. 19, translated line by line using the interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.7-2 20852. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
16  ( ps  \/  ( ch  /\  th ) )  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) ) Given $e. No need for adantr 451 because we do not move this into an ND hypothesis
21 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption (new scope) simpr 447
32 ...  ( ps  \/  ch )  ( ( ph  /\  ps )  ->  ( ps  \/  ch ) )  \/IL 2 orcd 381, the MPE equivalent of  \/IL, 1
43 ...|  ( ch  /\  th )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ch  /\  th ) ) ND hypothesis assumption (new scope) simpr 447
54 ...  ch  ( ( ph  /\  ( ch  /\  th ) )  ->  ch )  /\EL 4 simpld 445, the MPE equivalent of  /\EL, 3
66 ...  ( ps  \/  ch )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ps  \/  ch ) )  \/IR 5 olcd 382, the MPE equivalent of  \/IR, 4
77  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) )  \/E 1,3,6 mpjaodan 761, the MPE equivalent of  \/E, 2,5,6

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  \/  ch )
 )
 
Theoremex-natded5.7-2 20852 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 20851. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  \/  ch )
 )
 
Theoremex-natded5.8 20853 Theorem 5.8 of [Clemente] p. 20, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
110;11  ( ( ps  /\  ch )  ->  -.  th )  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) ) Given $e; adantr 451 to move it into the ND hypothesis
23;4  ( ta  ->  th )  ( ph  ->  ( ta  ->  th ) ) Given $e; adantr 451 to move it into the ND hypothesis
37;8  ch  ( ph  ->  ch ) Given $e; adantr 451 to move it into the ND hypothesis
41;2  ta  ( ph  ->  ta ) Given $e. adantr 451 to move it into the ND hypothesis
56 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND Hypothesis/Assumption simpr 447. New ND hypothesis scope, each reference outside the scope must change antedent  ph to  ( ph  /\  ps ).
69 ...  ( ps  /\  ch )  ( ( ph  /\  ps )  ->  ( ps  /\  ch ) )  /\I 5,3 jca 518 ( /\I), 6,8 (adantr 451 to bring in scope)
75 ...  -.  th  ( ( ph  /\  ps )  ->  -.  th )  ->E 1,6 mpd 14 ( ->E), 2,4
812 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 14 ( ->E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913  -.  ps  ( ph  ->  -.  ps )  -.I 5,7,8 pm2.65da 559 ( -.I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 451; simpr 447 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.8-2 20854.

(Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  -.  th )
 )   &    |-  ( ph  ->  ( ta  ->  th ) )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ta )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.8-2 20854 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 20853. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  -.  th )
 )   &    |-  ( ph  ->  ( ta  ->  th ) )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ta )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.13 20855 Theorem 5.13 of [Clemente] p. 20, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 20856. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
115  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) ) Given $e.
2;32  ( ps  ->  th )  ( ph  ->  ( ps  ->  th ) ) Given $e. adantr 451 to move it into the ND hypothesis
39  ( -.  ta  ->  -.  ch )  ( ph  ->  ( -.  ta  ->  -.  ch ) ) Given $e. ad2antrr 706 to move it into the ND sub-hypothesis
41 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 447
54 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 14 1,3
65 ...  ( th  \/  ta )  ( ( ph  /\  ps )  ->  ( th  \/  ta ) )  \/I 5 orcd 381 4
76 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 447
88 ... ...|  -.  ta  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta ) (sub) ND hypothesis assumption simpr 447
911 ... ...  -.  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )  ->E 3,8 mpd 14 8,10
107 ... ...  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch ) IT 7 adantr 451 6
1112 ...  -.  -.  ta  ( ( ph  /\  ch )  ->  -.  -.  ta )  -.I 8,9,10 pm2.65da 559 7,11
1213 ...  ta  ( ( ph  /\  ch )  ->  ta )  -.E 11 notnotrd 105 12
1314 ...  ( th  \/  ta )  ( ( ph  /\  ch )  ->  ( th  \/  ta ) )  \/I 12 olcd 382 13
1416  ( th  \/  ta )  ( ph  ->  ( th  \/  ta ) )  \/E 1,6,13 mpjaodan 761 5,14,15

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 451; simpr 447 is useful when you want to depend directly on the new assumption). (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( -.  ta  ->  -.  ch ) )   =>    |-  ( ph  ->  ( th  \/  ta ) )
 
Theoremex-natded5.13-2 20856 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 20855. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( -.  ta  ->  -.  ch ) )   =>    |-  ( ph  ->  ( th  \/  ta ) )
 
Theoremex-natded9.20 20857 Theorem 9.20 of [Clemente] p. 43, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
11  ( ps  /\  ( ch  \/  th ) )  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) ) Given $e
22  ps  ( ph  ->  ps )  /\EL 1 simpld 445 1
311  ( ch  \/  th )  ( ph  ->  ( ch  \/  th ) )  /\ER 1 simprd 449 1
44 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 447
55 ...  ( ps  /\  ch )  ( ( ph  /\  ch )  ->  ( ps  /\  ch ) )  /\I 2,4 jca 518 3,4
66 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IR 5 orcd 381 5
78 ...|  th  ( ( ph  /\  th )  ->  th ) ND hypothesis assumption simpr 447
89 ...  ( ps  /\  th )  ( ( ph  /\  th )  ->  ( ps  /\  th ) )  /\I 2,7 jca 518 7,8
910 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IL 8 olcd 382 9
1012  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/E 3,6,9 mpjaodan 761 6,10,11

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 451; simpr 447 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is ex-natded9.20-2 20858. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

 |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
 
Theoremex-natded9.20-2 20858 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 20857. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)
 |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
 
Theoremex-natded9.26 20859* Theorem 9.26 of [Clemente] p. 45, translated line by line using an interpretation of natural deduction in Metamath. This proof has some additional complications due to the fact that Metamath's existential elimination rule does not change bound variables, so we need to verify that  x is bound in the conclusion. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
13  E. x A. y ps ( x ,  y )  ( ph  ->  E. x A. y ps ) Given $e.
26 ...|  A. y ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  A. y ps ) ND hypothesis assumption simpr 447. Later statements will have this scope.
37;5,4 ...  ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  ps )  A.E 2,y spsbcd 3038 ( A.E), 5,6. To use it we need a1i 10 and vex 2825. This could be immediately done with 19.21bi 1825, but we want to show the general approach for substitution.
412;8,9,10,11 ...  E. x ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  E. x ps )  E.I 3,a spesbcd 3107 ( E.I), 11. To use it we need sylibr 203, which in turn requires sylib 188 and two uses of sbcid 3041. This could be more immediately done using 19.8a 1739, but we want to show the general approach for substitution.
513;1,2  E. x ps ( x ,  y )  ( ph  ->  E. x ps )  E.E 1,2,4,a exlimdd 1861 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1610 and nfe1 1723 (MPE# 1,2)
614  A. y E. x ps ( x ,  y )  ( ph  ->  A. y E. x ps )  A.I 5 alrimiv 1622 ( A.I), 13

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps).

Note that in the original proof,  ps ( x ,  y ) has explicit parameters. In Metamath, these parameters are always implicit, and the parameters upon which a wff variable can depend are recorded in the "allowed substitution hints" below.

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded9.26-2 20860.

(Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by David A. Wheeler, 18-Feb-2017.)

 |-  ( ph  ->  E. x A. y ps )   =>    |-  ( ph  ->  A. y E. x ps )
 
Theoremex-natded9.26-2 20860* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 20859. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  E. x A. y ps )   =>    |-  ( ph  ->  A. y E. x ps )
 
14.1.4  Definitional examples
 
Theoremex-or 20861 Example for df-or 359. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ( 2  =  3  \/  4  =  4 )
 
Theoremex-an 20862 Example for df-an 360. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ( 2  =  2 
 /\  3  =  3 )
 
Theoremex-dif 20863 Example for df-dif 3189. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  \  {
 1 ,  8 } )  =  { 3 }
 
Theoremex-un 20864 Example for df-un 3191. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  u.  {
 1 ,  8 } )  =  { 1 ,  3 ,  8 }
 
Theoremex-in 20865 Example for df-in 3193. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  i^i  {
 1 ,  8 } )  =  { 1 }
 
Theoremex-uni 20866 Example for df-uni 3865. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |- 
 U. { { 1 ,  3 } ,  { 1 ,  8 } }  =  {
 1 ,  3 ,  8 }
 
Theoremex-ss 20867 Example for df-ss 3200. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |- 
 { 1 ,  2 }  C_  { 1 ,  2 ,  3 }
 
Theoremex-pss 20868 Example for df-pss 3202. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |- 
 { 1 ,  2 }  C.  { 1 ,  2 ,  3 }
 
Theoremex-pw 20869 Example for df-pw 3661. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |-  ( A  =  {
 3 ,  5 ,  7 }  ->  ~P A  =  ( ( { (/) }  u.  { { 3 } ,  { 5 } ,  { 7 } }
 )  u.  ( { { 3 ,  5 } ,  { 3 ,  7 } ,  { 5 ,  7 } }  u.  { { 3 ,  5 ,  7 } }
 ) ) )
 
Theoremex-pr 20870 Example for df-pr 3681. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( A  e.  {
 1 ,  -u 1 }  ->  ( A ^
 2 )  =  1 )
 
Theoremex-br 20871 Example for df-br 4061. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( R  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  3 R
 9 )
 
Theoremex-opab 20872* Example for df-opab 4115. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( R  =  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1
 )  =  y ) }  ->  3 R
 4 )
 
Theoremex-eprel 20873 Example for df-eprel 4342. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  5  _E  { 1 ,  5 }
 
Theoremex-id 20874 Example for df-id 4346. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 5  _I  5  /\  -.  4  _I  5
 )
 
Theoremex-po 20875 Example for df-po 4351. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  (  <  Po  RR  /\ 
 -.  <_  Po  RR )
 
Theoremex-xp 20876 Example for df-xp 4732. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( { 1 ,  5 }  X.  {
 2 ,  7 } )  =  ( { <. 1 ,  2 >. ,  <. 1 ,  7
 >. }  u.  { <. 5 ,  2 >. ,  <. 5 ,  7 >. } )
 
Theoremex-cnv 20877 Example for df-cnv 4734. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  `' { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  =  { <. 6 ,  2 >. ,  <. 9 ,  3 >. }
 
Theoremex-co 20878 Example for df-co 4735. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( exp  o.  cos ) `  0 )  =  _e
 
Theoremex-dm 20879 Example for df-dm 4736. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  dom  F  =  { 2 ,  3 } )
 
Theoremex-rn 20880 Example for df-rn 4737. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  ran  F  =  { 6 ,  9 } )
 
Theoremex-res 20881 Example for df-res 4738. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( F  =  { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F  |`  B )  =  { <. 2 ,  6
 >. } )
 
Theoremex-ima 20882 Example for df-ima 4739. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( F  =  { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F " B )  =  { 6 } )
 
Theoremex-fv 20883 Example for df-fv 5300. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  ( F `  3 )  =  9 )
 
Theoremex-1st 20884 Example for df-1st 6164. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 1st `  <. 3 ,  4 >. )  =  3
 
Theoremex-2nd 20885 Example for df-2nd 6165. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 2nd `  <. 3 ,  4 >. )  =  4
 
Theorem1kp2ke3k 20886 Example for df-dec 10172, 1000 + 2000 = 3000.

This proof disproves (by counter-example) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with  ( 2  +  1 )  =  3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 10172 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

 |-  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0
 
Theoremex-fl 20887 Example for df-fl 10972. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ( |_ `  (
 3  /  2 )
 )  =  1  /\  ( |_ `  -u (
 3  /  2 )
 )  =  -u 2
 )
 
Theoremex-dvds 20888 3 divides into 6. A demonstration of df-dvds 12579. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  3  ||  6
 
14.2  Humor
 
14.2.1  April Fool's theorem
 
Theoremavril1 20889 Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid german dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object  x equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

A reply to skeptics can be found at http://us.metamath.org/mpeuni/mmnotes.txt, under the 1-Apr-2006 entry.

 |- 
 -.  ( A ~P RR ( _i `  1
 )  /\  F (/) ( 0  x.  1 ) )
 
Theorem2bornot2b 20890 The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 2  x.  B  \/  -.  2  x.  B )
 
Theoremhelloworld 20891 The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 -.  ( h  e.  ( L L 0 )  /\  W (/) ( R. 1 d ) )
 
Theorem1p1e2apr1 20892 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1  +  1 )  =  2
 
Theoremeqid1 20893 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). It is one of the three axioms of Ayn Rand's philosophy (Atlas Shrugged, Part Three, Chapter VII). While some have proposed extending Rand's axiomatization to include Compassion and Kindness, others fear that such an extension may flirt with logical inconsistency. (Contributed by Stefan Allan, 1-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  A  =  A
 
Theorem1div0apr 20894 Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1  /  0
 )  =  (/)
 
14.3  (Future - to be reviewed and classified)
 
14.3.1  Planar incidence geometry
 
Syntaxcplig 20895 Extend class notation with the class of all planar incidence geometries.
 class  Plig
 
Definitiondf-plig 20896* Planar incidence geometry. I use Hilbert's "axioms" adapted to planar geometry.  e. is the incidence relation. I could take a generic incidence relation but I'm lazy and I'm not sure the gain is worth the extra work. Much of what follows is directly borrowed from Aitken. http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf (Contributed by FL, 2-Aug-2009.)
 |- 
 Plig  =  { x  |  ( A. a  e. 
 U. x A. b  e.  U. x ( a  =/=  b  ->  E! l  e.  x  (
 a  e.  l  /\  b  e.  l )
 )  /\  A. l  e.  x  E. a  e. 
 U. x E. b  e.  U. x ( a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  U. x E. b  e.  U. x E. c  e.  U. x A. l  e.  x  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l
 ) ) }
 
Theoremisplig 20897* The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)
 |-  P  =  U. L   =>    |-  ( L  e.  A  ->  ( L  e.  Plig  <->  ( A. a  e.  P  A. b  e.  P  ( a  =/=  b  ->  E! l  e.  L  ( a  e.  l  /\  b  e.  l ) )  /\  A. l  e.  L  E. a  e.  P  E. b  e.  P  ( a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) ) ) )
 
Theoremtncp 20898* There exist three non colinear points. (Contributed by FL, 3-Aug-2009.)
 |-  P  =  U. L   =>    |-  ( L  e.  Plig  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) )
 
Theoremlpni 20899* For any line, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)
 |-  P  =  U. G   =>    |-  (
 ( G  e.  Plig  /\  L  e.  G ) 
 ->  E. a  e.  P  a  e/  L )
 
14.3.2  Algebra preliminaries
 
Syntaxcrpm 20900 Ring primes.
 class RPrime
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