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Theorem pntlemq 20750
Description: Lemma for pntlemj 20752. (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 20746 . . . . . . . . 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 20744 . . . . . . . . . 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 10875 . . . . . . . . . . 11  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ZZ )
2220, 21syl 15 . . . . . . . . . 10  |-  ( ph  ->  J  e.  ZZ )
2322peano2zd 10120 . . . . . . . . 9  |-  ( ph  ->  ( J  +  1 )  e.  ZZ )
2419, 23rpexpcld 11268 . . . . . . . 8  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  e.  RR+ )
2517, 24rpdivcld 10407 . . . . . . 7  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR+ )
2625rpred 10390 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR )
2726flcld 10930 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  e.  ZZ )
28 1rp 10358 . . . . . . . . . 10  |-  1  e.  RR+
291, 2, 3, 4, 5, 6pntlemd 20743 . . . . . . . . . . . 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 10406 . . . . . . . . . 10  |-  ( ph  ->  ( L  x.  E
)  e.  RR+ )
33 rpaddcl 10374 . . . . . . . . . 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 10406 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  e.  RR+ )
3717, 36rpdivcld 10407 . . . . . . 7  |-  ( ph  ->  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR+ )
3837rpred 10390 . . . . . 6  |-  ( ph  ->  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR )
3938flcld 10930 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ZZ )
4036rpred 10390 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  e.  RR )
4124rpred 10390 . . . . . . . 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 10392 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  CC )
4619, 22rpexpcld 11268 . . . . . . . . . . . 12  |-  ( ph  ->  ( K ^ J
)  e.  RR+ )
4746rpcnd 10392 . . . . . . . . . . 11  |-  ( ph  ->  ( K ^ J
)  e.  CC )
4845, 47mulcomd 8856 . . . . . . . . . 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 20747 . . . . . . . . . . . . . 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 10879 . . . . . . . . . . . . . 14  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ( ZZ>= `  M )
)
5420, 53syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  ( ZZ>= `  M ) )
55 nnuz 10263 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
5655uztrn2 10245 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  J  e.  ( ZZ>= `  M ) )  ->  J  e.  NN )
5752, 54, 56syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  NN )
5857nnnn0d 10018 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  NN0 )
5945, 58expp1d 11246 . . . . . . . . . 10  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  =  ( ( K ^ J )  x.  K ) )
6048, 59eqtr4d 2318 . . . . . . . . 9  |-  ( ph  ->  ( K  x.  ( K ^ J ) )  =  ( K ^
( J  +  1 ) ) )
6144, 60breqtrd 4047 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K ^ ( J  + 
1 ) ) )
6240, 41, 61ltled 8967 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <_  ( K ^ ( J  + 
1 ) ) )
6336, 24, 17lediv2d 10414 . . . . . . 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 10942 . . . . . 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 10236 . . . . 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 10253 . . . 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 10830 . . . 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 10407 . . . . . . 7  |-  ( ph  ->  ( Z  /  V
)  e.  RR+ )
7372rpred 10390 . . . . . 6  |-  ( ph  ->  ( Z  /  V
)  e.  RR )
7473flcld 10930 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  V ) )  e.  ZZ )
7517, 46rpdivcld 10407 . . . . . . 7  |-  ( ph  ->  ( Z  /  ( K ^ J ) )  e.  RR+ )
7675rpred 10390 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ J ) )  e.  RR )
7776flcld 10930 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ZZ )
7846rpred 10390 . . . . . . . 8  |-  ( ph  ->  ( K ^ J
)  e.  RR )
7935rpred 10390 . . . . . . . 8  |-  ( ph  ->  V  e.  RR )
8043simpld 445 . . . . . . . 8  |-  ( ph  ->  ( K ^ J
)  <  V )
8178, 79, 80ltled 8967 . . . . . . 7  |-  ( ph  ->  ( K ^ J
)  <_  V )
8246, 35, 17lediv2d 10414 . . . . . . 7  |-  ( ph  ->  ( ( K ^ J )  <_  V  <->  ( Z  /  V )  <_  ( Z  / 
( K ^ J
) ) ) )
8381, 82mpbid 201 . . . . . 6  |-  ( ph  ->  ( Z  /  V
)  <_  ( Z  /  ( K ^ J ) ) )
84 flwordi 10942 . . . . . 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 10236 . . . . 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 10831 . . . 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 3189 . 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 3219 1  |-  ( ph  ->  I  C_  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   3c3 9796   4c4 9797   ZZcz 10024  ;cdc 10124   ZZ>=cuz 10230   RR+crp 10354   (,)cioo 10656   [,)cico 10658   [,]cicc 10659   ...cfz 10782  ..^cfzo 10870   |_cfl 10924   ^cexp 11104   sqrcsqr 11718   abscabs 11719   expce 12343   _eceu 12344   logclog 19912  ψcchp 20330
This theorem is referenced by:  pntlemj  20752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-e 12350  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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