MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pntlemq Unicode version

Theorem pntlemq 20766
Description: Lemma for pntlemj 20768. (Contributed by Mario Carneiro, 7-Jun-2016.)
Hypotheses
Ref Expression
pntlem1.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
pntlem1.a  |-  ( ph  ->  A  e.  RR+ )
pntlem1.b  |-  ( ph  ->  B  e.  RR+ )
pntlem1.l  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
pntlem1.d  |-  D  =  ( A  +  1 )
pntlem1.f  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
pntlem1.u  |-  ( ph  ->  U  e.  RR+ )
pntlem1.u2  |-  ( ph  ->  U  <_  A )
pntlem1.e  |-  E  =  ( U  /  D
)
pntlem1.k  |-  K  =  ( exp `  ( B  /  E ) )
pntlem1.y  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
pntlem1.x  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
pntlem1.c  |-  ( ph  ->  C  e.  RR+ )
pntlem1.w  |-  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
) ) ) ) )
pntlem1.z  |-  ( ph  ->  Z  e.  ( W [,)  +oo ) )
pntlem1.m  |-  M  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )
pntlem1.n  |-  N  =  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
pntlem1.U  |-  ( ph  ->  A. z  e.  ( Y [,)  +oo )
( abs `  (
( R `  z
)  /  z ) )  <_  U )
pntlem1.K  |-  ( 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 ) )
pntlem1.o  |-  O  =  ( ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  / 
( K ^ J
) ) ) )
pntlem1.v  |-  ( ph  ->  V  e.  RR+ )
pntlem1.V  |-  ( 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 )
)
pntlem1.j  |-  ( ph  ->  J  e.  ( M..^ N ) )
pntlem1.i  |-  I  =  ( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) )
Assertion
Ref Expression
pntlemq  |-  ( ph  ->  I  C_  O )
Distinct variable groups:    z, C    y, z, J    y, u, z, L    y, K, z   
z, M    z, O    z, N    u, R, y, z    u, V    z, U    z, W    y, X, z    z, Y    u, a,
y, z, E    u, Z, z
Allowed substitution hints:    ph( y, z, u, a)    A( y, z, u, a)    B( y, z, u, a)    C( y, u, a)    D( y, z, u, a)    R( a)    U( y, u, a)    F( y, z, u, a)    I( y, z, u, a)    J( u, a)    K( u, a)    L( a)    M( y, u, a)    N( y, u, a)    O( y, u, a)    V( y, z, a)    W( y, u, a)    X( u, a)    Y( y, u, a)    Z( y, a)

Proof of Theorem pntlemq
StepHypRef Expression
1 pntlem1.r . . . . . . . . . 10  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
2 pntlem1.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR+ )
3 pntlem1.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR+ )
4 pntlem1.l . . . . . . . . . 10  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
5 pntlem1.d . . . . . . . . . 10  |-  D  =  ( A  +  1 )
6 pntlem1.f . . . . . . . . . 10  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
7 pntlem1.u . . . . . . . . . 10  |-  ( ph  ->  U  e.  RR+ )
8 pntlem1.u2 . . . . . . . . . 10  |-  ( ph  ->  U  <_  A )
9 pntlem1.e . . . . . . . . . 10  |-  E  =  ( U  /  D
)
10 pntlem1.k . . . . . . . . . 10  |-  K  =  ( exp `  ( B  /  E ) )
11 pntlem1.y . . . . . . . . . 10  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
12 pntlem1.x . . . . . . . . . 10  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
13 pntlem1.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  RR+ )
14 pntlem1.w . . . . . . . . . 10  |-  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
) ) ) ) )
15 pntlem1.z . . . . . . . . . 10  |-  ( ph  ->  Z  e.  ( W [,)  +oo ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15pntlemb 20762 . . . . . . . . 9  |-  ( ph  ->  ( Z  e.  RR+  /\  ( 1  <  Z  /\  _e  <_  ( sqr `  Z )  /\  ( sqr `  Z )  <_ 
( Z  /  Y
) )  /\  (
( 4  /  ( L  x.  E )
)  <_  ( sqr `  Z )  /\  (
( ( log `  X
)  /  ( log `  K ) )  +  2 )  <_  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  /\  (
( U  x.  3 )  +  C )  <_  ( ( ( U  -  E )  x.  ( ( L  x.  ( E ^
2 ) )  / 
(; 3 2  x.  B
) ) )  x.  ( log `  Z
) ) ) ) )
1716simp1d 967 . . . . . . . 8  |-  ( ph  ->  Z  e.  RR+ )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10pntlemc 20760 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1918simp2d 968 . . . . . . . . 9  |-  ( ph  ->  K  e.  RR+ )
20 pntlem1.j . . . . . . . . . . 11  |-  ( ph  ->  J  e.  ( M..^ N ) )
21 elfzoelz 10891 . . . . . . . . . . 11  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ZZ )
2220, 21syl 15 . . . . . . . . . 10  |-  ( ph  ->  J  e.  ZZ )
2322peano2zd 10136 . . . . . . . . 9  |-  ( ph  ->  ( J  +  1 )  e.  ZZ )
2419, 23rpexpcld 11284 . . . . . . . 8  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  e.  RR+ )
2517, 24rpdivcld 10423 . . . . . . 7  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR+ )
2625rpred 10406 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR )
2726flcld 10946 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  e.  ZZ )
28 1rp 10374 . . . . . . . . . 10  |-  1  e.  RR+
291, 2, 3, 4, 5, 6pntlemd 20759 . . . . . . . . . . . 12  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
3029simp1d 967 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  RR+ )
3118simp1d 967 . . . . . . . . . . 11  |-  ( ph  ->  E  e.  RR+ )
3230, 31rpmulcld 10422 . . . . . . . . . 10  |-  ( ph  ->  ( L  x.  E
)  e.  RR+ )
33 rpaddcl 10390 . . . . . . . . . 10  |-  ( ( 1  e.  RR+  /\  ( L  x.  E )  e.  RR+ )  ->  (
1  +  ( L  x.  E ) )  e.  RR+ )
3428, 32, 33sylancr 644 . . . . . . . . 9  |-  ( ph  ->  ( 1  +  ( L  x.  E ) )  e.  RR+ )
35 pntlem1.v . . . . . . . . 9  |-  ( ph  ->  V  e.  RR+ )
3634, 35rpmulcld 10422 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  e.  RR+ )
3717, 36rpdivcld 10423 . . . . . . 7  |-  ( ph  ->  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR+ )
3837rpred 10406 . . . . . 6  |-  ( ph  ->  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR )
3938flcld 10946 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ZZ )
4036rpred 10406 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  e.  RR )
4124rpred 10406 . . . . . . . 8  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  e.  RR )
42 pntlem1.V . . . . . . . . . . 11  |-  ( 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 )
)
4342simpld 445 . . . . . . . . . 10  |-  ( ph  ->  ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K  x.  ( K ^ J
) ) ) )
4443simprd 449 . . . . . . . . 9  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K  x.  ( K ^ J
) ) )
4519rpcnd 10408 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  CC )
4619, 22rpexpcld 11284 . . . . . . . . . . . 12  |-  ( ph  ->  ( K ^ J
)  e.  RR+ )
4746rpcnd 10408 . . . . . . . . . . 11  |-  ( ph  ->  ( K ^ J
)  e.  CC )
4845, 47mulcomd 8872 . . . . . . . . . 10  |-  ( ph  ->  ( K  x.  ( K ^ J ) )  =  ( ( K ^ J )  x.  K ) )
49 pntlem1.m . . . . . . . . . . . . . . 15  |-  M  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )
50 pntlem1.n . . . . . . . . . . . . . . 15  |-  N  =  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
511, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 49, 50pntlemg 20763 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
5251simp1d 967 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  NN )
53 elfzouz 10895 . . . . . . . . . . . . . 14  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ( ZZ>= `  M )
)
5420, 53syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  ( ZZ>= `  M ) )
55 nnuz 10279 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
5655uztrn2 10261 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  J  e.  ( ZZ>= `  M ) )  ->  J  e.  NN )
5752, 54, 56syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  NN )
5857nnnn0d 10034 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  NN0 )
5945, 58expp1d 11262 . . . . . . . . . 10  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  =  ( ( K ^ J )  x.  K ) )
6048, 59eqtr4d 2331 . . . . . . . . 9  |-  ( ph  ->  ( K  x.  ( K ^ J ) )  =  ( K ^
( J  +  1 ) ) )
6144, 60breqtrd 4063 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K ^ ( J  + 
1 ) ) )
6240, 41, 61ltled 8983 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <_  ( K ^ ( J  + 
1 ) ) )
6336, 24, 17lediv2d 10430 . . . . . . 7  |-  ( ph  ->  ( ( ( 1  +  ( L  x.  E ) )  x.  V )  <_  ( K ^ ( J  + 
1 ) )  <->  ( Z  /  ( K ^
( J  +  1 ) ) )  <_ 
( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) ) ) )
6462, 63mpbid 201 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  <_  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )
65 flwordi 10958 . . . . . 6  |-  ( ( ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR  /\  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR  /\  ( Z  /  ( K ^
( J  +  1 ) ) )  <_ 
( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) ) )  ->  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  <_ 
( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ) )
6626, 38, 64, 65syl3anc 1182 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  <_  ( |_ `  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) ) ) )
67 eluz2 10252 . . . . 5  |-  ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) ) )  <-> 
( ( |_ `  ( Z  /  ( K ^ ( J  + 
1 ) ) ) )  e.  ZZ  /\  ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ZZ  /\  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  <_ 
( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ) ) )
6827, 39, 66, 67syl3anbrc 1136 . . . 4  |-  ( ph  ->  ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) ) ) )
69 eluzp1p1 10269 . . . 4  |-  ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) ) )  ->  ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 )  e.  (
ZZ>= `  ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ) )
70 fzss1 10846 . . . 4  |-  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 )  e.  ( ZZ>= `  ( ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) )  -> 
( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) 
C_  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) )
7168, 69, 703syl 18 . . 3  |-  ( ph  ->  ( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) 
C_  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) )
7217, 35rpdivcld 10423 . . . . . . 7  |-  ( ph  ->  ( Z  /  V
)  e.  RR+ )
7372rpred 10406 . . . . . 6  |-  ( ph  ->  ( Z  /  V
)  e.  RR )
7473flcld 10946 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  V ) )  e.  ZZ )
7517, 46rpdivcld 10423 . . . . . . 7  |-  ( ph  ->  ( Z  /  ( K ^ J ) )  e.  RR+ )
7675rpred 10406 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ J ) )  e.  RR )
7776flcld 10946 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ZZ )
7846rpred 10406 . . . . . . . 8  |-  ( ph  ->  ( K ^ J
)  e.  RR )
7935rpred 10406 . . . . . . . 8  |-  ( ph  ->  V  e.  RR )
8043simpld 445 . . . . . . . 8  |-  ( ph  ->  ( K ^ J
)  <  V )
8178, 79, 80ltled 8983 . . . . . . 7  |-  ( ph  ->  ( K ^ J
)  <_  V )
8246, 35, 17lediv2d 10430 . . . . . . 7  |-  ( ph  ->  ( ( K ^ J )  <_  V  <->  ( Z  /  V )  <_  ( Z  / 
( K ^ J
) ) ) )
8381, 82mpbid 201 . . . . . 6  |-  ( ph  ->  ( Z  /  V
)  <_  ( Z  /  ( K ^ J ) ) )
84 flwordi 10958 . . . . . 6  |-  ( ( ( Z  /  V
)  e.  RR  /\  ( Z  /  ( K ^ J ) )  e.  RR  /\  ( Z  /  V )  <_ 
( Z  /  ( K ^ J ) ) )  ->  ( |_ `  ( Z  /  V
) )  <_  ( |_ `  ( Z  / 
( K ^ J
) ) ) )
8573, 76, 83, 84syl3anc 1182 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  V ) )  <_  ( |_ `  ( Z  /  ( K ^ J ) ) ) )
86 eluz2 10252 . . . . 5  |-  ( ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  /  V ) ) )  <-> 
( ( |_ `  ( Z  /  V
) )  e.  ZZ  /\  ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ZZ  /\  ( |_ `  ( Z  /  V ) )  <_ 
( |_ `  ( Z  /  ( K ^ J ) ) ) ) )
8774, 77, 85, 86syl3anbrc 1136 . . . 4  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  /  V ) ) ) )
88 fzss2 10847 . . . 4  |-  ( ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  /  V ) ) )  ->  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) )  C_  ( (
( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  ( K ^ J ) ) ) ) )
8987, 88syl 15 . . 3  |-  ( ph  ->  ( ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) 
C_  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  ( K ^ J ) ) ) ) )
9071, 89sstrd 3202 . 2  |-  ( ph  ->  ( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) 
C_  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  ( K ^ J ) ) ) ) )
91 pntlem1.i . 2  |-  I  =  ( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) )
92 pntlem1.o . 2  |-  O  =  ( ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  / 
( K ^ J
) ) ) )
9390, 91, 923sstr4g 3232 1  |-  ( ph  ->  I  C_  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    +oocpnf 8880    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   3c3 9812   4c4 9813   ZZcz 10040  ;cdc 10140   ZZ>=cuz 10246   RR+crp 10370   (,)cioo 10672   [,)cico 10674   [,]cicc 10675   ...cfz 10798  ..^cfzo 10886   |_cfl 10940   ^cexp 11120   sqrcsqr 11734   abscabs 11735   expce 12359   _eceu 12360   logclog 19928  ψcchp 20346
This theorem is referenced by:  pntlemj  20768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-e 12366  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930
  Copyright terms: Public domain W3C validator