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Theorem pntlemh 21285
Description: Lemma for pnt 21300. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-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 ) )
Assertion
Ref Expression
pntlemh  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
Distinct variable group:    E, a
Allowed substitution hints:    ph( a)    A( a)    B( a)    C( a)    D( a)    R( a)    U( a)    F( a)    J( a)    K( a)    L( a)    M( a)    N( a)    W( a)    X( a)    Y( a)    Z( a)

Proof of Theorem pntlemh
StepHypRef Expression
1 pntlem1.x . . . . . . . . . 10  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
21simpld 446 . . . . . . . . 9  |-  ( ph  ->  X  e.  RR+ )
32adantr 452 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  X  e.  RR+ )
43relogcld 20510 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  X )  e.  RR )
5 pntlem1.r . . . . . . . . . . . 12  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
6 pntlem1.a . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR+ )
7 pntlem1.b . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  RR+ )
8 pntlem1.l . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
9 pntlem1.d . . . . . . . . . . . 12  |-  D  =  ( A  +  1 )
10 pntlem1.f . . . . . . . . . . . 12  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
11 pntlem1.u . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  RR+ )
12 pntlem1.u2 . . . . . . . . . . . 12  |-  ( ph  ->  U  <_  A )
13 pntlem1.e . . . . . . . . . . . 12  |-  E  =  ( U  /  D
)
14 pntlem1.k . . . . . . . . . . . 12  |-  K  =  ( exp `  ( B  /  E ) )
155, 6, 7, 8, 9, 10, 11, 12, 13, 14pntlemc 21281 . . . . . . . . . . 11  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1615simp2d 970 . . . . . . . . . 10  |-  ( ph  ->  K  e.  RR+ )
1716rpred 10640 . . . . . . . . 9  |-  ( ph  ->  K  e.  RR )
1815simp3d 971 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
1918simp2d 970 . . . . . . . . 9  |-  ( ph  ->  1  <  K )
2017, 19rplogcld 20516 . . . . . . . 8  |-  ( ph  ->  ( log `  K
)  e.  RR+ )
2120adantr 452 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR+ )
224, 21rerpdivcld 10667 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  e.  RR )
23 pntlem1.y . . . . . . . . . 10  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
24 pntlem1.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  RR+ )
25 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
) ) ) ) )
26 pntlem1.z . . . . . . . . . 10  |-  ( ph  ->  Z  e.  ( W [,)  +oo ) )
27 pntlem1.m . . . . . . . . . 10  |-  M  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )
28 pntlem1.n . . . . . . . . . 10  |-  N  =  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
295, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23, 1, 24, 25, 26, 27, 28pntlemg 21284 . . . . . . . . 9  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
3029simp1d 969 . . . . . . . 8  |-  ( ph  ->  M  e.  NN )
3130adantr 452 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  e.  NN )
3231nnred 10007 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  e.  RR )
33 elfzuz 11047 . . . . . . . 8  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
34 nnuz 10513 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
3534uztrn2 10495 . . . . . . . 8  |-  ( ( M  e.  NN  /\  J  e.  ( ZZ>= `  M ) )  ->  J  e.  NN )
3630, 33, 35syl2an 464 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  NN )
3736nnred 10007 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  RR )
38 flltp1 11201 . . . . . . . 8  |-  ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  ->  ( ( log `  X )  / 
( log `  K
) )  <  (
( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 ) )
3922, 38syl 16 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  (
( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 ) )
4039, 27syl6breqr 4244 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  M
)
41 elfzle1 11052 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  M  <_  J )
4241adantl 453 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  <_  J )
4322, 32, 37, 40, 42ltletrd 9222 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  J
)
444, 37, 21ltdivmul2d 10688 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( log `  X
)  /  ( log `  K ) )  < 
J  <->  ( log `  X
)  <  ( J  x.  ( log `  K
) ) ) )
4543, 44mpbid 202 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  X )  <  ( J  x.  ( log `  K ) ) )
4616adantr 452 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  K  e.  RR+ )
47 elfzelz 11051 . . . . . 6  |-  ( J  e.  ( M ... N )  ->  J  e.  ZZ )
4847adantl 453 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  ZZ )
49 relogexp 20482 . . . . 5  |-  ( ( K  e.  RR+  /\  J  e.  ZZ )  ->  ( log `  ( K ^ J ) )  =  ( J  x.  ( log `  K ) ) )
5046, 48, 49syl2anc 643 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( K ^ J
) )  =  ( J  x.  ( log `  K ) ) )
5145, 50breqtrrd 4230 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  X )  <  ( log `  ( K ^ J ) ) )
5246, 48rpexpcld 11538 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( K ^ J )  e.  RR+ )
53 logltb 20486 . . . 4  |-  ( ( X  e.  RR+  /\  ( K ^ J )  e.  RR+ )  ->  ( X  <  ( K ^ J )  <->  ( log `  X )  <  ( log `  ( K ^ J ) ) ) )
543, 52, 53syl2anc 643 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  <->  ( log `  X
)  <  ( log `  ( K ^ J
) ) ) )
5551, 54mpbird 224 . 2  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  X  <  ( K ^ J ) )
5650oveq2d 6089 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  ( log `  ( K ^ J ) ) )  =  ( 2  x.  ( J  x.  ( log `  K ) ) ) )
57 2z 10304 . . . . . . . 8  |-  2  e.  ZZ
58 relogexp 20482 . . . . . . . 8  |-  ( ( ( K ^ J
)  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( ( K ^ J ) ^
2 ) )  =  ( 2  x.  ( log `  ( K ^ J ) ) ) )
5952, 57, 58sylancl 644 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  =  ( 2  x.  ( log `  ( K ^ J
) ) ) )
60 2cn 10062 . . . . . . . . 9  |-  2  e.  CC
6160a1i 11 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  CC )
6237recnd 9106 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  CC )
6346relogcld 20510 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR )
6463recnd 9106 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  CC )
6561, 62, 64mulassd 9103 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  x.  ( log `  K
) )  =  ( 2  x.  ( J  x.  ( log `  K
) ) ) )
6656, 59, 653eqtr4d 2477 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  =  ( ( 2  x.  J
)  x.  ( log `  K ) ) )
67 elfzle2 11053 . . . . . . . . . . 11  |-  ( J  e.  ( M ... N )  ->  J  <_  N )
6867adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  N )
6968, 28syl6breq 4243 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  ( |_ `  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) ) )
705, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23, 1, 24, 25, 26pntlemb 21283 . . . . . . . . . . . . . . 15  |-  ( 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
) ) ) ) )
7170simp1d 969 . . . . . . . . . . . . . 14  |-  ( ph  ->  Z  e.  RR+ )
7271adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  Z  e.  RR+ )
7372relogcld 20510 . . . . . . . . . . . 12  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  Z )  e.  RR )
7473, 21rerpdivcld 10667 . . . . . . . . . . 11  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  Z )  / 
( log `  K
) )  e.  RR )
7574rehalfcld 10206 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( log `  Z
)  /  ( log `  K ) )  / 
2 )  e.  RR )
76 flge 11206 . . . . . . . . . 10  |-  ( ( ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  e.  RR  /\  J  e.  ZZ )  ->  ( J  <_  ( ( ( log `  Z )  /  ( log `  K
) )  /  2
)  <->  J  <_  ( |_
`  ( ( ( log `  Z )  /  ( log `  K
) )  /  2
) ) ) )
7775, 48, 76syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( J  <_  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <->  J  <_  ( |_ `  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) ) ) )
7869, 77mpbird 224 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
79 2re 10061 . . . . . . . . . 10  |-  2  e.  RR
8079a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  RR )
81 2pos 10074 . . . . . . . . . 10  |-  0  <  2
8281a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  0  <  2 )
83 lemuldiv2 9882 . . . . . . . . 9  |-  ( ( J  e.  RR  /\  ( ( log `  Z
)  /  ( log `  K ) )  e.  RR  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
( ( 2  x.  J )  <_  (
( log `  Z
)  /  ( log `  K ) )  <->  J  <_  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) ) )
8437, 74, 80, 82, 83syl112anc 1188 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  <_  ( ( log `  Z )  /  ( log `  K ) )  <-> 
J  <_  ( (
( log `  Z
)  /  ( log `  K ) )  / 
2 ) ) )
8578, 84mpbird 224 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  J )  <_ 
( ( log `  Z
)  /  ( log `  K ) ) )
86 remulcl 9067 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  J  e.  RR )  ->  ( 2  x.  J
)  e.  RR )
8779, 37, 86sylancr 645 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  J )  e.  RR )
8887, 73, 21lemuldivd 10685 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( 2  x.  J
)  x.  ( log `  K ) )  <_ 
( log `  Z
)  <->  ( 2  x.  J )  <_  (
( log `  Z
)  /  ( log `  K ) ) ) )
8985, 88mpbird 224 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  x.  ( log `  K
) )  <_  ( log `  Z ) )
9066, 89eqbrtrd 4224 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  <_  ( log `  Z ) )
91 rpexpcl 11392 . . . . . . 7  |-  ( ( ( K ^ J
)  e.  RR+  /\  2  e.  ZZ )  ->  (
( K ^ J
) ^ 2 )  e.  RR+ )
9252, 57, 91sylancl 644 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  e.  RR+ )
9392, 72logled 20514 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( K ^ J
) ^ 2 )  <_  Z  <->  ( log `  ( ( K ^ J ) ^ 2 ) )  <_  ( log `  Z ) ) )
9490, 93mpbird 224 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  <_  Z
)
9572rprege0d 10647 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( Z  e.  RR  /\  0  <_  Z ) )
96 resqrth 12053 . . . . 5  |-  ( ( Z  e.  RR  /\  0  <_  Z )  -> 
( ( sqr `  Z
) ^ 2 )  =  Z )
9795, 96syl 16 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( sqr `  Z ) ^
2 )  =  Z )
9894, 97breqtrrd 4230 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  <_  (
( sqr `  Z
) ^ 2 ) )
9952rprege0d 10647 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J )  e.  RR  /\  0  <_ 
( K ^ J
) ) )
10072rpsqrcld 12206 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( sqr `  Z )  e.  RR+ )
101100rprege0d 10647 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( sqr `  Z )  e.  RR  /\  0  <_ 
( sqr `  Z
) ) )
102 le2sq 11448 . . . 4  |-  ( ( ( ( K ^ J )  e.  RR  /\  0  <_  ( K ^ J ) )  /\  ( ( sqr `  Z
)  e.  RR  /\  0  <_  ( sqr `  Z
) ) )  -> 
( ( K ^ J )  <_  ( sqr `  Z )  <->  ( ( K ^ J ) ^
2 )  <_  (
( sqr `  Z
) ^ 2 ) ) )
10399, 101, 102syl2anc 643 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J )  <_ 
( sqr `  Z
)  <->  ( ( K ^ J ) ^
2 )  <_  (
( sqr `  Z
) ^ 2 ) ) )
10498, 103mpbird 224 . 2  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( K ^ J )  <_  ( sqr `  Z ) )
10555, 104jca 519 1  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    +oocpnf 9109    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   3c3 10042   4c4 10043   ZZcz 10274  ;cdc 10374   ZZ>=cuz 10480   RR+crp 10604   (,)cioo 10908   [,)cico 10910   ...cfz 11035   |_cfl 11193   ^cexp 11374   sqrcsqr 12030   expce 12656   _eceu 12657   logclog 20444  ψcchp 20867
This theorem is referenced by:  pntlemr  21288  pntlemj  21289  pntlemi  21290  pntlemf  21291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-e 12663  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446
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