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Theorem List for Metamath Proof Explorer - 27801-27900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstoweidlem52 27801* There exists a neighborood V as in Lemma 1 of [BrosowskiDeutsh] p. 90. Here Z is used to represent t0 in the paper, and v is used to represent V in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  F/_ t P   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  V  =  { t  e.  T  |  ( P `  t
 )  <  ( D  /  2 ) }   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  a  e.  RR )  ->  (
 t  e.  T  |->  a )  e.  A )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  D  <  1 )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t ) 
 <_  1 ) )   &    |-  ( ph  ->  ( P `  Z )  =  0
 )   &    |-  ( ph  ->  A. t  e.  ( T  \  U ) D  <_  ( P `
  t ) )   =>    |-  ( ph  ->  E. v  e.  J  ( ( Z  e.  v  /\  v  C_  U )  /\  A. e  e.  RR+  E. x  e.  A  ( A. t  e.  T  ( 0  <_  ( x `  t ) 
 /\  ( x `  t )  <_  1 ) 
 /\  A. t  e.  v  ( x `  t )  <  e  /\  A. t  e.  ( T  \  U ) ( 1  -  e )  < 
 ( x `  t
 ) ) ) )
 
Theoremstoweidlem53 27802* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  W  =  { w  e.  J  |  E. h  e.  Q  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } }   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  J  e.  Comp
 )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  ( T  \  U )  =/=  (/) )   &    |-  ( ph  ->  Z  e.  U )   =>    |-  ( ph  ->  E. p  e.  A  (
 A. t  e.  T  ( 0  <_  ( p `  t )  /\  ( p `  t ) 
 <_  1 )  /\  ( p `  Z )  =  0  /\  A. t  e.  ( T  \  U ) 0  <  ( p `  t ) ) )
 
Theoremstoweidlem54 27803* There exists a function  x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here  D is used to represent  A in the paper, because here  A is used for the subalgebra of functions.  E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ i ph   &    |-  F/ t ph   &    |-  F/ y ph   &    |-  F/ w ph   &    |-  T  =  U. J   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) }   &    |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) ) )   &    |-  F  =  ( t  e.  T  |->  ( i  e.  (
 1 ... M )  |->  ( ( y `  i
 ) `  t )
 ) )   &    |-  Z  =  ( t  e.  T  |->  ( 
 seq  1 (  x. 
 ,  ( F `  t ) ) `  M ) )   &    |-  V  =  { w  e.  J  |  A. e  e.  RR+  E. h  e.  A  (
 A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 )  /\  A. t  e.  w  ( h `  t )  < 
 e  /\  A. t  e.  ( T  \  U ) ( 1  -  e )  <  ( h `
  t ) ) }   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  W : ( 1 ...
 M ) --> V )   &    |-  ( ph  ->  B  C_  T )   &    |-  ( ph  ->  D  C_ 
 U. ran  W )   &    |-  ( ph  ->  D  C_  T )   &    |-  ( ph  ->  E. y
 ( y : ( 1 ... M ) --> Y  /\  A. i  e.  ( 1 ... M ) ( A. t  e.  ( W `  i
 ) ( ( y `
  i ) `  t )  <  ( E 
 /  M )  /\  A. t  e.  B  ( 1  -  ( E 
 /  M ) )  <  ( ( y `
  i ) `  t ) ) ) )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  /  3
 ) )   =>    |-  ( ph  ->  E. x  e.  A  ( A. t  e.  T  ( 0  <_  ( x `  t ) 
 /\  ( x `  t )  <_  1 ) 
 /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  (
 1  -  E )  <  ( x `  t ) ) )
 
Theoremstoweidlem55 27804* This lemma proves the existence of a function p as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  Z  e.  U )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  W  =  { w  e.  J  |  E. h  e.  Q  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } }   =>    |-  ( ph  ->  E. p  e.  A  (
 A. t  e.  T  ( 0  <_  ( p `  t )  /\  ( p `  t ) 
 <_  1 )  /\  ( p `  Z )  =  0  /\  A. t  e.  ( T  \  U ) 0  <  ( p `  t ) ) )
 
Theoremstoweidlem56 27805* This theorem proves Lemma 1 in [BrosowskiDeutsh] p. 90. Here  Z is used to represent t0 in the paper,  v is used to represent  V in the paper, and  e is used to represent ε (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  y  e.  RR )  ->  (
 t  e.  T  |->  y )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  Z  e.  U )   =>    |-  ( ph  ->  E. v  e.  J  ( ( Z  e.  v  /\  v  C_  U )  /\  A. e  e.  RR+  E. x  e.  A  ( A. t  e.  T  ( 0  <_  ( x `  t ) 
 /\  ( x `  t )  <_  1 ) 
 /\  A. t  e.  v  ( x `  t )  <  e  /\  A. t  e.  ( T  \  U ) ( 1  -  e )  < 
 ( x `  t
 ) ) ) )
 
Theoremstoweidlem57 27806* There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. In this theorem, it is proven the non trivial case (the closed set D is nonempty). Here D is used to represent A in the paper, because the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t D   &    |-  F/_ t U   &    |-  F/ t ph   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) }   &    |-  V  =  { w  e.  J  |  A. e  e.  RR+  E. h  e.  A  (
 A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 )  /\  A. t  e.  w  ( h `  t )  < 
 e  /\  A. t  e.  ( T  \  U ) ( 1  -  e )  <  ( h `
  t ) ) }   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  U  =  ( T  \  B )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  a  e.  RR )  ->  (
 t  e.  T  |->  a )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  B  e.  ( Clsd `  J ) )   &    |-  ( ph  ->  D  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  ( B  i^i  D )  =  (/) )   &    |-  ( ph  ->  D  =/=  (/) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. x  e.  A  (
 A. t  e.  T  ( 0  <_  ( x `  t )  /\  ( x `  t ) 
 <_  1 )  /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  ( 1  -  E )  <  ( x `
  t ) ) )
 
Theoremstoweidlem58 27807* This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 91. Here D is used to represent the set A of Lemma 2, because here the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t D   &    |-  F/_ t U   &    |-  F/ t ph   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  J  e.  Comp
 )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  a  e.  RR )  ->  (
 t  e.  T  |->  a )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  B  e.  ( Clsd `  J ) )   &    |-  ( ph  ->  D  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  ( B  i^i  D )  =  (/) )   &    |-  U  =  ( T  \  B )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. x  e.  A  (
 A. t  e.  T  ( 0  <_  ( x `  t )  /\  ( x `  t ) 
 <_  1 )  /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  ( 1  -  E )  <  ( x `
  t ) ) )
 
Theoremstoweidlem59 27808* This lemma proves that there exists a function  x as in the proof in [BrosowskiDeutsh] p. 91, after Lemma 2: xj is in the subalgebra, 0 <= xj <= 1, xj < ε / n on Aj (meaning A in the paper), xj > 1 - \epslon / n on Bj. Here  D is used to represent A in the paper (because A is used for the subalgebra of functions),  E is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  D  =  ( j  e.  ( 0 ... N )  |->  { t  e.  T  |  ( F `  t
 )  <_  ( (
 j  -  ( 1 
 /  3 ) )  x.  E ) }
 )   &    |-  B  =  ( j  e.  ( 0 ...
 N )  |->  { t  e.  T  |  ( ( j  +  ( 1 
 /  3 ) )  x.  E )  <_  ( F `  t ) } )   &    |-  Y  =  {
 y  e.  A  |  A. t  e.  T  ( 0  <_  (
 y `  t )  /\  ( y `  t
 )  <_  1 ) }   &    |-  H  =  ( j  e.  ( 0 ...
 N )  |->  { y  e.  Y  |  ( A. t  e.  ( D `  j ) ( y `
  t )  < 
 ( E  /  N )  /\  A. t  e.  ( B `  j
 ) ( 1  -  ( E  /  N ) )  <  ( y `
  t ) ) } )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  y  e.  RR )  ->  (
 t  e.  T  |->  y )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  E. x ( x : ( 0
 ... N ) --> A  /\  A. j  e.  ( 0
 ... N ) (
 A. t  e.  T  ( 0  <_  (
 ( x `  j
 ) `  t )  /\  ( ( x `  j ) `  t
 )  <_  1 )  /\  A. t  e.  ( D `  j ) ( ( x `  j
 ) `  t )  <  ( E  /  N )  /\  A. t  e.  ( B `  j
 ) ( 1  -  ( E  /  N ) )  <  ( ( x `  j ) `
  t ) ) ) )
 
Theoremstoweidlem60 27809* This lemma proves that there exists a function g as in the proof in [BrosowskiDeutsh] p. 91 (this parte of the proof actually spans through pages 91-92): g is in the subalgebra, and for all  t in  T, there is a  j such that (j-4/3)*ε < f(t) <= (j-1/3)*ε and (j-4/3)*ε < g(t) < (j+1/3)*ε. Here  F is used to represent f in the paper, and  E is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  D  =  ( j  e.  ( 0 ... n )  |->  { t  e.  T  |  ( F `  t
 )  <_  ( (
 j  -  ( 1 
 /  3 ) )  x.  E ) }
 )   &    |-  B  =  ( j  e.  ( 0 ... n )  |->  { t  e.  T  |  ( ( j  +  ( 1 
 /  3 ) )  x.  E )  <_  ( F `  t ) } )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  T  =/=  (/) )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  y  e.  RR )  ->  (
 t  e.  T  |->  y )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  A. t  e.  T  0  <_  ( F `  t ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. g  e.  A  A. t  e.  T  E. j  e.  RR  ( ( ( ( j  -  (
 4  /  3 )
 )  x.  E )  <  ( F `  t )  /\  ( F `
  t )  <_  ( ( j  -  ( 1  /  3
 ) )  x.  E ) )  /\  ( ( g `  t )  <  ( ( j  +  ( 1  / 
 3 ) )  x.  E )  /\  (
 ( j  -  (
 4  /  3 )
 )  x.  E )  <  ( g `  t ) ) ) )
 
Theoremstoweidlem61 27810* This lemma proves that there exists a function  g as in the proof in [BrosowskiDeutsh] p. 92:  g is in the subalgebra, and for all  t in  T, abs( f(t) - g(t) ) < 2*ε. Here  F is used to represent f in the paper, and  E is used to represent ε. For this lemma there's the further assumption that the function  F to be approximated is nonnegative (this assumption is removed in a later theorem). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  T  =  U. J   &    |-  ( ph  ->  T  =/=  (/) )   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  A. t  e.  T  0  <_  ( F `  t ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. g  e.  A  A. t  e.  T  ( abs `  ( ( g `
  t )  -  ( F `  t ) ) )  <  (
 2  x.  E ) )
 
Theoremstoweidlem62 27811* This theorem proves the Stone Weierstrass theorem for the non trivial case in which T is nonempty. The proof follows [BrosowskiDeutsh] p. 89 (through page 92). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ f ph   &    |-  F/ t ph   &    |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  ( ph  ->  J  e.  Comp )   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  T  =/=  (/) )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `
  t )  -  ( F `  t ) ) )  <  E )
 
Theoremstoweid 27812* This theorem proves the Stone-Weierstrass theorem for real valued functions: let  J be a compact topology on  T, and  C be the set of real continuous functions on  T. Assume that  A is a subalgebra of  C (closed under addition and multiplication of functions) containing constant functions and discriminating points (if  r and  t are distinct points in  T, then there exists a function  h in  A such that h(r) is distinct from h(t) ). Then, for any continuous function 
F and for any positive real  E, there exists a function  f in the subalgebra  A, such that  f approximates  F up to  E ( E represents the usual ε value). As a classical example, given any a,b reals, the closed interval  T  =  [
a ,  b ] could be taken, along with the subalgebra  A of real polynomials on  T, and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on  [ a ,  b ]. The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. h  e.  A  ( h `  r )  =/=  ( h `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  E  e.  RR+ )   =>    |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  (
 ( f `  t
 )  -  ( F `
  t ) ) )  <  E )
 
Theoremstowei 27813* This theorem proves the Stone-Weierstrass theorem for real valued functions: let  J be a compact topology on  T, and  C be the set of real continuous functions on  T. Assume that  A is a subalgebra of  C (closed under addition and multiplication of functions) containing constant functions and discriminating points (if  r and  t are distinct points in  T, then there exists a function  h in  A such that h(r) is distinct from h(t) ). Then, for any continuous function 
F and for any positive real  E, there exists a function  f in the subalgebra  A, such that  f approximates  F up to  E ( E represents the usual ε value). As a classical example, given any a,b reals, the closed interval  T  =  [
a ,  b ] could be taken, along with the subalgebra  A of real polynomials on  T, and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on  [ a ,  b ]. The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. The deduction version of this theorem is stoweid 27812: often times it will be better to use stoweid 27812 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  K  =  ( topGen `  ran  (,) )   &    |-  J  e.  Comp   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  A  C_  C   &    |-  ( ( f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( x  e.  RR  ->  ( t  e.  T  |->  x )  e.  A )   &    |-  ( ( r  e.  T  /\  t  e.  T  /\  r  =/=  t )  ->  E. h  e.  A  ( h `  r )  =/=  ( h `  t ) )   &    |-  F  e.  C   &    |-  E  e.  RR+   =>    |-  E. f  e.  A  A. t  e.  T  ( abs `  (
 ( f `  t
 )  -  ( F `
  t ) ) )  <  E
 
18.20.7  Wallis' product for π
 
Theoremwallispilem1 27814*  I is monotone: increasing the exponent, the integral decreases. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  I  =  ( n  e.  NN0  |->  S. ( 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( I `  ( N  +  1 ) )  <_  ( I `  N ) )
 
Theoremwallispilem2 27815* A first set of properties for the sequence  I that will be used in the proof of the Wallis product formula (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  I  =  ( n  e.  NN0  |->  S. ( 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   =>    |-  ( ( I `
  0 )  =  pi  /\  ( I `
  1 )  =  2  /\  ( N  e.  ( ZZ>= `  2
 )  ->  ( I `  N )  =  ( ( ( N  -  1 )  /  N )  x.  ( I `  ( N  -  2
 ) ) ) ) )
 
Theoremwallispilem3 27816* I maps to real values (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  I  =  ( n  e.  NN0  |->  S. ( 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   =>    |-  ( N  e.  NN0 
 ->  ( I `  N )  e.  RR+ )
 
Theoremwallispilem4 27817*  F maps to explicit expression for the ratio of two consecutive values of  I (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  F  =  ( k  e.  NN  |->  ( ( ( 2  x.  k )  /  ( ( 2  x.  k )  -  1
 ) )  x.  (
 ( 2  x.  k
 )  /  ( (
 2  x.  k )  +  1 ) ) ) )   &    |-  I  =  ( n  e.  NN0  |->  S. (
 0 (,) pi ) ( ( sin `  z
 ) ^ n )  _d z )   &    |-  G  =  ( n  e.  NN  |->  ( ( I `  ( 2  x.  n ) )  /  ( I `  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( ( pi  /  2
 )  x.  ( 1 
 /  (  seq  1
 (  x.  ,  F ) `  n ) ) ) )   =>    |-  G  =  H
 
Theoremwallispilem5 27818* The sequence  H converges to 1. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  F  =  ( k  e.  NN  |->  ( ( ( 2  x.  k )  /  ( ( 2  x.  k )  -  1
 ) )  x.  (
 ( 2  x.  k
 )  /  ( (
 2  x.  k )  +  1 ) ) ) )   &    |-  I  =  ( n  e.  NN0  |->  S. (
 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   &    |-  G  =  ( n  e.  NN  |->  ( ( I `  (
 2  x.  n ) )  /  ( I `
  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( ( pi  /  2
 )  x.  ( 1 
 /  (  seq  1
 (  x.  ,  F ) `  n ) ) ) )   &    |-  L  =  ( n  e.  NN  |->  ( ( ( 2  x.  n )  +  1 )  /  ( 2  x.  n ) ) )   =>    |-  H  ~~>  1
 
Theoremwallispi 27819* Wallis' formula for π : Wallis' product converges to π / 2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F  =  ( k  e.  NN  |->  ( ( ( 2  x.  k )  /  ( ( 2  x.  k )  -  1
 ) )  x.  (
 ( 2  x.  k
 )  /  ( (
 2  x.  k )  +  1 ) ) ) )   &    |-  W  =  ( n  e.  NN  |->  ( 
 seq  1 (  x. 
 ,  F ) `  n ) )   =>    |-  W  ~~>  ( pi  /  2 )
 
Theoremwallispi2lem1 27820 An intermediate step between the first version of the Wallis' formula for π and the second version of Wallis' formula. This second version will then be used to prove Stirling's approximation formula for the factorial. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  ( N  e.  NN  ->  ( 
 seq  1 (  x. 
 ,  ( k  e. 
 NN  |->  ( ( ( 2  x.  k ) 
 /  ( ( 2  x.  k )  -  1 ) )  x.  ( ( 2  x.  k )  /  (
 ( 2  x.  k
 )  +  1 ) ) ) ) ) `
  N )  =  ( ( 1  /  ( ( 2  x.  N )  +  1 ) )  x.  (  seq  1 (  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
 4 )  /  (
 ( ( 2  x.  k )  x.  (
 ( 2  x.  k
 )  -  1 ) ) ^ 2 ) ) ) ) `  N ) ) )
 
Theoremwallispi2lem2 27821 Two expressions are proven to be equal, and this is used to complete the proof of the second version of Wallis' formula for π . (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  ( N  e.  NN  ->  ( 
 seq  1 (  x. 
 ,  ( k  e. 
 NN  |->  ( ( ( 2  x.  k ) ^ 4 )  /  ( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  -  1
 ) ) ^ 2
 ) ) ) ) `
  N )  =  ( ( ( 2 ^ ( 4  x.  N ) )  x.  ( ( ! `  N ) ^ 4
 ) )  /  (
 ( ! `  (
 2  x.  N ) ) ^ 2 ) ) )
 
Theoremwallispi2 27822 An alternative version of Wallis' formula for π ; this second formula uses factorials and it is later used to proof Stirling's approximation formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
  n ) ^
 4 ) )  /  ( ( ! `  ( 2  x.  n ) ) ^ 2
 ) )  /  (
 ( 2  x.  n )  +  1 )
 ) )   =>    |-  V  ~~>  ( pi  / 
 2 )
 
18.20.8  Stirling's approximation formula for ` n ` factorial
 
Theoremstirlinglem1 27823 A simple limit of fractions is computed. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  H  =  ( n  e.  NN  |->  ( ( n ^
 2 )  /  ( n  x.  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  F  =  ( n  e.  NN  |->  ( 1  -  ( 1 
 /  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  G  =  ( n  e.  NN  |->  ( 1  /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  L  =  ( n  e.  NN  |->  ( 1  /  n ) )   =>    |-  H  ~~>  ( 1  / 
 2 )
 
Theoremstirlinglem2 27824  A maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   =>    |-  ( N  e.  NN  ->  ( A `  N )  e.  RR+ )
 
Theoremstirlinglem3 27825 Long but simple algebraic transformations are applied to show that  V, the Wallis formula for π , can be expressed in terms of  A, the Stirling's approximation formula for the factorial, up to a constant factor. This will allow (in a later theorem) to determine the right constant factor to be put into the  A, in order to get the exact Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  D  =  ( n  e.  NN  |->  ( A `  ( 2  x.  n ) ) )   &    |-  E  =  ( n  e.  NN  |->  ( ( sqr `  (
 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) )   &    |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `  n ) ^ 4
 ) )  /  (
 ( ! `  (
 2  x.  n ) ) ^ 2 ) )  /  ( ( 2  x.  n )  +  1 ) ) )   =>    |-  V  =  ( n  e.  NN  |->  ( ( ( ( A `  n ) ^ 4
 )  /  ( ( D `  n ) ^
 2 ) )  x.  ( ( n ^
 2 )  /  ( n  x.  ( ( 2  x.  n )  +  1 ) ) ) ) )
 
Theoremstirlinglem4 27826* Algebraic manipulation of  ( ( B n ) - ( B  ( n  +  1 ) ) ). It will be used in other theorems to show that  B is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  J  =  ( n  e.  NN  |->  ( ( ( ( 1  +  (
 2  x.  n ) )  /  2 )  x.  ( log `  (
 ( n  +  1 )  /  n ) ) )  -  1
 ) )   =>    |-  ( N  e.  NN  ->  ( ( B `  N )  -  ( B `  ( N  +  1 ) ) )  =  ( J `  N ) )
 
Theoremstirlinglem5 27827* If  T is between  0 and  1, then a series (without alternating negative and positive terms) is given that converges to log (1+T)/(1-T) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  D  =  ( j  e.  NN  |->  ( ( -u 1 ^ ( j  -  1 ) )  x.  ( ( T ^
 j )  /  j
 ) ) )   &    |-  E  =  ( j  e.  NN  |->  ( ( T ^
 j )  /  j
 ) )   &    |-  F  =  ( j  e.  NN  |->  ( ( ( -u 1 ^ ( j  -  1 ) )  x.  ( ( T ^
 j )  /  j
 ) )  +  (
 ( T ^ j
 )  /  j )
 ) )   &    |-  H  =  ( j  e.  NN0  |->  ( 2  x.  ( ( 1 
 /  ( ( 2  x.  j )  +  1 ) )  x.  ( T ^ (
 ( 2  x.  j
 )  +  1 ) ) ) ) )   &    |-  G  =  ( j  e.  NN0  |->  ( ( 2  x.  j )  +  1 ) )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  ( abs `  T )  < 
 1 )   =>    |-  ( ph  ->  seq  0
 (  +  ,  H ) 
 ~~>  ( log `  (
 ( 1  +  T )  /  ( 1  -  T ) ) ) )
 
Theoremstirlinglem6 27828* A series that converges to log (N+1)/N (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  H  =  ( j  e.  NN0  |->  ( 2  x.  (
 ( 1  /  (
 ( 2  x.  j
 )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ ( ( 2  x.  j )  +  1 ) ) ) ) )   =>    |-  ( N  e.  NN  ->  seq  0 (  +  ,  H )  ~~>  ( log `  ( ( N  +  1 )  /  N ) ) )
 
Theoremstirlinglem7 27829* Algebraic manipulation of the formula for J(n) (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  J  =  ( n  e.  NN  |->  ( ( ( ( 1  +  ( 2  x.  n ) ) 
 /  2 )  x.  ( log `  (
 ( n  +  1 )  /  n ) ) )  -  1
 ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1  /  (
 ( 2  x.  k
 )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ ( 2  x.  k ) ) ) )   &    |-  H  =  ( k  e.  NN0  |->  ( 2  x.  ( ( 1 
 /  ( ( 2  x.  k )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ (
 ( 2  x.  k
 )  +  1 ) ) ) ) )   =>    |-  ( N  e.  NN  ->  seq  1 (  +  ,  K )  ~~>  ( J `  N ) )
 
Theoremstirlinglem8 27830 If  A converges to  C, then  F converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ n ph   &    |-  F/_ n A   &    |-  F/_ n D   &    |-  D  =  ( n  e.  NN  |->  ( A `
  ( 2  x.  n ) ) )   &    |-  ( ph  ->  A : NN
 --> RR+ )   &    |-  F  =  ( n  e.  NN  |->  ( ( ( A `  n ) ^ 4
 )  /  ( ( D `  n ) ^
 2 ) ) )   &    |-  L  =  ( n  e.  NN  |->  ( ( A `
  n ) ^
 4 ) )   &    |-  M  =  ( n  e.  NN  |->  ( ( D `  n ) ^ 2
 ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( D `  n )  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  ~~>  C )   =>    |-  ( ph  ->  F  ~~>  ( C ^ 2 ) )
 
Theoremstirlinglem9 27831*  ( ( B `  N )  -  ( B `  ( N  +  1
) ) ) is expressed as a limit of a series. This result will be used both to prove that  B is decreasing and to prove that  B is bounded (below). It will follow that  B converges in the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  J  =  ( n  e.  NN  |->  ( ( ( ( 1  +  (
 2  x.  n ) )  /  2 )  x.  ( log `  (
 ( n  +  1 )  /  n ) ) )  -  1
 ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1  /  (
 ( 2  x.  k
 )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ ( 2  x.  k ) ) ) )   =>    |-  ( N  e.  NN  ->  seq  1 (  +  ,  K )  ~~>  ( ( B `  N )  -  ( B `  ( N  +  1 ) ) ) )
 
Theoremstirlinglem10 27832* A bound for any B(N)-B(N + 1) that will allow to find a lower bound for the whole  B sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1 
 /  ( ( 2  x.  k )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ (
 2  x.  k ) ) ) )   &    |-  L  =  ( k  e.  NN  |->  ( ( 1  /  ( ( ( 2  x.  N )  +  1 ) ^ 2
 ) ) ^ k
 ) )   =>    |-  ( N  e.  NN  ->  ( ( B `  N )  -  ( B `  ( N  +  1 ) ) ) 
 <_  ( ( 1  / 
 4 )  x.  (
 1  /  ( N  x.  ( N  +  1 ) ) ) ) )
 
Theoremstirlinglem11 27833*  B is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1 
 /  ( ( 2  x.  k )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ (
 2  x.  k ) ) ) )   =>    |-  ( N  e.  NN  ->  ( B `  ( N  +  1
 ) )  <  ( B `  N ) )
 
Theoremstirlinglem12 27834* The sequence  B is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  F  =  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  +  1 )
 ) ) )   =>    |-  ( N  e.  NN  ->  ( ( B `
  1 )  -  ( 1  /  4
 ) )  <_  ( B `  N ) )
 
Theoremstirlinglem13 27835*  B is decreasing and has a lower bound, then it converges. Since  B is  log A, in another theorem it is proven that  A converges also. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   =>    |-  E. d  e.  RR  B  ~~>  d
 
Theoremstirlinglem14 27836* The sequence  A converges to a positive real. This proves that the Stirling's formula converges to the factorial, up to a constant. In another theorem, using Wallis' formula for π& , such constant is exactly determined, thus proving the Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   =>    |-  E. c  e.  RR+  A  ~~>  c
 
Theoremstirlinglem15 27837* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 27838 will use this final lemma, but it will not expose the local definitions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ n ph   &    |-  S  =  ( n  e.  NN0  |->  ( ( sqr `  ( (
 2  x.  pi )  x.  n ) )  x.  ( ( n 
 /  _e ) ^ n ) ) )   &    |-  A  =  ( n  e.  NN  |->  ( ( ! `
  n )  /  ( ( sqr `  (
 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  D  =  ( n  e.  NN  |->  ( A `  ( 2  x.  n ) ) )   &    |-  E  =  ( n  e.  NN  |->  ( ( sqr `  (
 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) )   &    |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `  n ) ^ 4
 ) )  /  (
 ( ! `  (
 2  x.  n ) ) ^ 2 ) )  /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  F  =  ( n  e.  NN  |->  ( ( ( A `  n ) ^ 4
 )  /  ( ( D `  n ) ^
 2 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( ( n ^ 2 )  /  ( n  x.  (
 ( 2  x.  n )  +  1 )
 ) ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  ~~>  C )   =>    |-  ( ph  ->  ( n  e.  NN  |->  ( ( ! `  n ) 
 /  ( S `  n ) ) )  ~~>  1 )
 
Theoremstirling 27838 Stirling's approximation formula for 
n factorial. The proof follows two major steps: first it is proven that  S and  n factorial are asymptotically equivalent, up to an unknown constant. Then, using Wallis' formula for π it is proven that the unknown constant is the square root of π and then the exact Stirling's formula is established. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  S  =  ( n  e.  NN0  |->  ( ( sqr `  (
 ( 2  x.  pi )  x.  n ) )  x.  ( ( n 
 /  _e ) ^ n ) ) )   =>    |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( S `  n ) ) )  ~~>  1
 
Theoremstirlingr 27839 Stirling's approximation formula for 
n factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 27838 is proven for convergence in the topology of complex numbers. The variable  R is used to denote convergence with respect to the standard topology on the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  S  =  ( n  e.  NN0  |->  ( ( sqr `  (
 ( 2  x.  pi )  x.  n ) )  x.  ( ( n 
 /  _e ) ^ n ) ) )   &    |-  R  =  ( ~~> t `  ( topGen `  ran  (,) )
 )   =>    |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( S `  n ) ) ) R 1
 
18.21  Mathbox for Saveliy Skresanov
 
18.21.1  Ceva's theorem
 
Theoremsigarval 27840* Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  ( Im `  ( ( * `
  A )  x.  B ) ) )
 
Theoremsigarim 27841* Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  e.  RR )
 
Theoremsigarac 27842* Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
 
Theoremsigaraf 27843* Signed area is additive by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  C ) G B )  =  ( ( A G B )  +  ( C G B ) ) )
 
Theoremsigarmf 27844* Signed area is additive (with respect to subtraction) by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C ) G B )  =  ( ( A G B )  -  ( C G B ) ) )
 
Theoremsigaras 27845* Signed area is additive by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  +  C ) )  =  ( ( A G B )  +  ( A G C ) ) )
 
Theoremsigarms 27846* Signed area is additive (with respect to subtraction) by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  -  C ) )  =  ( ( A G B )  -  ( A G C ) ) )
 
Theoremsigarls 27847* Signed area is linear by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  ( A G ( B  x.  C ) )  =  ( ( A G B )  x.  C ) )
 
Theoremsigarid 27848* Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( A  e.  CC  ->  ( A G A )  =  0 )
 
Theoremsigarexp 27849* Expand the signed area formula by linearity. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( ( A G B )  -  ( A G C ) )  -  ( C G B ) ) )
 
Theoremsigarperm 27850* Signed area  ( A  -  C ) G ( B  -  C ) acts as a double area of a triangle  A B C. Here we prove that cyclically permuting the vertices doesn't change the area. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( B  -  A ) G ( C  -  A ) ) )
 
Theoremsigardiv 27851* If signed area between vectors  B  -  A and  C  -  A is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  -.  C  =  A )   &    |-  ( ph  ->  ( ( B  -  A ) G ( C  -  A ) )  =  0 )   =>    |-  ( ph  ->  (
 ( B  -  A )  /  ( C  -  A ) )  e. 
 RR )
 
Theoremsigarimcd 27852* Signed area takes value in complex numbers. Deduction version. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC )
 )   =>    |-  ( ph  ->  ( A G B )  e. 
 CC )
 
Theoremsigariz 27853* If signed area is zero, the signed area with swapped arguments is also zero. Deduction version. ( Contributed by Saveliy Skresanov, 23-Sep-2017.) (Contributed by Saveliy Skresanov, 24-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC )
 )   &    |-  ( ph  ->  ( A G B )  =  0 )   =>    |-  ( ph  ->  ( B G A )  =  0 )
 
Theoremsigarcol 27854* Given three points  A,  B and  C such that  -.  A  =  B, the point  C lies on the line going through  A and  B iff the corresponding signed area is zero. That justifies the usage of signed area as a collinearity indicator. (Contributed by Saveliy Skresanov, 22-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  -.  A  =  B )   =>    |-  ( ph  ->  (
 ( ( A  -  C ) G ( B  -  C ) )  =  0  <->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) ) )
 
Theoremsharhght 27855* Let  A B C be a triangle, and let  D lie on the line  A B. Then (doubled) areas of triangles  A D C and  C D B relate as lengths of corresponding bases  A D and  D B. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0
 ) )   =>    |-  ( ph  ->  (
 ( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) ) )
 
Theoremsigaradd 27856* Subtracting (double) area of  A D C from  A B C yields the (double) area of  D B C. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0
 ) )   =>    |-  ( ph  ->  (
 ( ( B  -  C ) G ( A  -  C ) )  -  ( ( D  -  C ) G ( A  -  C ) ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
 
Theoremcevathlem1 27857 Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
 |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
 )   &    |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC ) )   &    |-  ( ph  ->  ( G  e.  CC  /\  H  e.  CC  /\  K  e.  CC ) )   &    |-  ( ph  ->  ( A  =/=  0  /\  E  =/=  0  /\  C  =/=  0 ) )   &    |-  ( ph  ->  ( ( A  x.  B )  =  ( C  x.  D )  /\  ( E  x.  F )  =  ( A  x.  G )  /\  ( C  x.  H )  =  ( E  x.  K ) ) )   =>    |-  ( ph  ->  (
 ( B  x.  F )  x.  H )  =  ( ( D  x.  G )  x.  K ) )
 
Theoremcevathlem2 27858* Ceva's theorem second lemma. Relate (doubled) areas of triangles  C A O and 
A B O with of segments  B D and 
D C. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
 )   &    |-  ( ph  ->  O  e.  CC )   &    |-  ( ph  ->  ( ( ( A  -  O ) G ( D  -  O ) )  =  0  /\  ( ( B  -  O ) G ( E  -  O ) )  =  0  /\  ( ( C  -  O ) G ( F  -  O ) )  =  0 ) )   &    |-  ( ph  ->  ( ( ( A  -  F ) G ( B  -  F ) )  =  0  /\  ( ( B  -  D ) G ( C  -  D ) )  =  0  /\  ( ( C  -  E ) G ( A  -  E ) )  =  0 ) )   &    |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( B  -  O ) G ( C  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )   =>    |-  ( ph  ->  (
 ( ( C  -  O ) G ( A  -  O ) )  x.  ( B  -  D ) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  ( D  -  C ) ) )
 
Theoremcevath 27859* Ceva's theorem. Let  A B C be a triangle and let points  F,  D and  E lie on sides  A B,  B C,  C A correspondingly. Suppose that cevians  A D,  B E and  C F intersect at one point  O. Then triangle's sides are partitioned into segments and their lengths satisfy a certain identity. Here we obtain a bit stronger version by using complex numbers themselves instead of their absolute values.

The proof goes by applying cevathlem2 27858 three times and then using cevathlem1 27857 to multiply obtained identities and prove the theorem.

In the theorem statement we are using function  G as a collinearity indicator. For justification of that use, see sigarcol 27854. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
 )   &    |-  ( ph  ->  O  e.  CC )   &    |-  ( ph  ->  ( ( ( A  -  O ) G ( D  -  O ) )  =  0  /\  ( ( B  -  O ) G ( E  -  O ) )  =  0  /\  ( ( C  -  O ) G ( F  -  O ) )  =  0 ) )   &    |-  ( ph  ->  ( ( ( A  -  F ) G ( B  -  F ) )  =  0  /\  ( ( B  -  D ) G ( C  -  D ) )  =  0  /\  ( ( C  -  E ) G ( A  -  E ) )  =  0 ) )   &    |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( B  -  O ) G ( C  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )   =>    |-  ( ph  ->  (
 ( ( A  -  F )  x.  ( C  -  E ) )  x.  ( B  -  D ) )  =  ( ( ( F  -  B )  x.  ( E  -  A ) )  x.  ( D  -  C ) ) )
 
18.22  Mathbox for Jarvin Udandy
 
TheoremhirstL-ax3 27860 The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.)
 |-  (
 ( -.  ph  ->  -. 
 ps )  ->  (
 ( -.  ph  ->  ps )  ->  ph ) )
 
Theoremax3h 27861 Recovery of ax-3 7 from hirstL-ax3 27860. (Contributed by Jarvin Udandy, 3-Jul-2015.)
 |-  (
 ( -.  ph  ->  -. 
 ps )  ->  ( ps  ->  ph ) )
 
Theoremaibandbiaiffaiffb 27862 A closed form showing (a implies b and b implies a) same-as (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  ->  ps )  /\  ( ps 
 ->  ph ) )  <->  ( ph  <->  ps ) )
 
Theoremaibandbiaiaiffb 27863 A closed form showing (a implies b and b implies a) implies (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  ->  ps )  /\  ( ps 
 ->  ph ) )  ->  ( ph  <->  ps ) )
 
Theoremnotatnand 27864 Do not use. Use intnanr instead. Given not a, there exists a proof for not (a and b). (Contributed by Jarvin Udandy, 31-Aug-2016.)
 |-  -.  ph   =>    |-  -.  ( ph  /\  ps )
 
Theoremaistia 27865 Given a is equivalent to T., there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
 |-  ( ph 
 <->  T.  )   =>    |-  ph
 
Theoremaisfina 27866 Given a is equivalent to F., there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
 |-  ( ph 
 <->  F.  )   =>    |- 
 -.  ph
 
Theorembothtbothsame 27867 Given both a,b are equivalent to T., there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
 |-  ( ph 
 <->  T.  )   &    |-  ( ps  <->  T.  )   =>    |-  ( ph  <->  ps )
 
Theorembothfbothsame 27868 Given both a,b are equivalent to F., there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  F.  )   =>    |-  ( ph  <->  ps )
 
Theoremaiffbbtat 27869 Given a is equivalent to b, b is equivalent to T. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)
 |-  ( ph 
 <->  ps )   &    |-  ( ps  <->  T.  )   =>    |-  ( ph  <->  T.  )
 
Theoremaisbbisfaisf 27870 Given a is equivalent to b, b is equivalent to F. there exists a proof for a is equivalent to F. (Contributed by Jarvin Udandy, 30-Aug-2016.)
 |-  ( ph 
 <->  ps )   &    |-  ( ps  <->  F.  )   =>    |-  ( ph  <->  F.  )
 
Theoremaxorbtnotaiffb 27871 Given a is exclusive to b, there exists a proof for (not (a if-and-only-if b)) df-xor is a closed form of this. (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ps )   =>    |-  -.  ( ph  <->  ps )
 
Theoremaiffnbandciffatnotciffb 27872 Given a is equivalent to NOT b, c is equivalent to a. there exists a proof for ( not ( c iff b ) ). (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph 
 <->  -.  ps )   &    |-  ( ch 
 <-> 
 ph )   =>    |- 
 -.  ( ch  <->  ps )
 
Theoremaxorbciffatcxorb 27873 Given a is equivalent to NOT b, c is equivalent to a. there exists a proof for ( c xor b ) . (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ps )   &    |-  ( ch 
 <-> 
 ph )   =>    |-  ( ch \/_ ps )
 
Theoremaibnbna 27874 Given a implies b, not b, there exists a proof for not a. (Contributed by Jarvin Udandy, 1-Sep-2016.)
 |-  ( ph  ->  ps )   &    |-  -.  ps   =>    |-  -.  ph
 
Theoremaibnbaif 27875 Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.)
 |-  ( ph  ->  ps )   &    |-  -.  ps   =>    |-  ( ph  <->  F.  )
 
Theoremaiffbtbat 27876 Given a is equivalent to b, T. is equivalent to b. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)
 |-  ( ph 
 <->  ps )   &    |-  (  T.  <->  ps )   =>    |-  ( ph  <->  T.  )
 
Theoremastbstanbst 27877 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for a and b is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)
 |-  ( ph 
 <->  T.  )   &    |-  ( ps  <->  T.  )   =>    |-  ( ( ph  /\ 
 ps )  <->  T.  )
 
Theoremaistbistaandb 27878 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for (a and b). (Contributed by Jarvin Udandy, 9-Sep-2016.)
 |-  ( ph 
 <->  T.  )   &    |-  ( ps  <->  T.  )   =>    |-  ( ph  /\  ps )
 
Theoremaisbnaxb 27879 Given a is equivalent to b, there exists a proof for (not (a xor b)). (Contributed by Jarvin Udandy, 28-Aug-2016.)
 |-  ( ph 
 <->  ps )   =>    |- 
 -.  ( ph \/_ ps )
 
Theoremiatbtatnnb 27880 Given a implies b, there exists a proof for a implies not not b. (Contributed by Jarvin Udandy, 2-Sep-2016.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  -.  -.  ps )
 
Theorematbiffatnnb 27881 If a implies b, is is implied a implies not not b (Contributed by Jarvin Udandy, 28-Aug-2016.)
 |-  (
 ( ph  ->  ps )  ->  ( ph  ->  -.  -.  ps ) )
 
Theorembisaiaisb 27882 Application of bicom1 with a, b swapped. (Contributed by Jarvin Udandy, 31-Aug-2016.)
 |-  (
 ( ps  <->  ph )  ->  ( ph 
 <->  ps ) )
 
Theorematbiffatnnbalt 27883 If a implies b, it is implied a implies not not b (Contributed by Jarvin Udandy, 29-Aug-2016.)
 |-  (
 ( ph  ->  ps )  ->  ( ph  ->  -.  -.  ps ) )
 
Theoremabnotbtaxb 27884 Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)
 |-  ph   &    |-  -.  ps   =>    |-  ( ph \/_ ps )
 
Theoremabnotataxb 27885 Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)
 |-  -.  ph   &    |-  ps   =>    |-  ( ph \/_ ps )
 
Theoremconimpf 27886 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.)
 |-  ph   &    |-  -.  ps   &    |-  ( ph  ->  ps )   =>    |-  ( ph  <->  F.  )
 
Theoremconimpfalt 27887 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 29-Aug-2016.)
 |-  ph   &    |-  -.  ps   &    |-  ( ph  ->  ps )   =>    |-  ( ph  <->  F.  )
 
Theoremaistbisfiaxb 27888 Given a is equivalent to T., Given b is equivalent to F. there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
 |-  ( ph 
 <->  T.  )   &    |-  ( ps  <->  F.  )   =>    |-  ( ph \/_ ps )
 
Theoremaisfbistiaxb 27889 Given a is equivalent to F., Given b is equivalent to T., there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   =>    |-  ( ph \/_ ps )
 
Theoremabcdta 27890 Given (((a and b) and c) and d), there exists a proof for a (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  /\  th )   =>    |-  ph
 
Theoremabcdtb 27891 Given (((a and b) and c) and d), there exists a proof for b (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  /\  th )   =>    |- 
 ps
 
Theoremabcdtc 27892 Given (((a and b) and c) and d), there exists a proof for c (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  /\  th )   =>    |- 
 ch
 
Theoremabcdtd 27893 Given (((a and b) and c) and d), there exists a proof for d (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  /\  th )   =>    |- 
 th
 
Theoremmdandyv0 27894 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  F.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv1 27895 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  F.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv2 27896 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  F.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv3 27897 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  F.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv4 27898 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  T.  )   &    |-  ( et 
 <->  F.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv5 27899 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  T.  )   &    |-  ( et 
 <->  F.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv6 27900 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  T.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ph ) )
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