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Theorem pntlemh 21246
Description: Lemma for pnt 21261. 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 20471 . . . . . . 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 21242 . . . . . . . . . . 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 10604 . . . . . . . . 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 20477 . . . . . . . 8  |-  ( ph  ->  ( log `  K
)  e.  RR+ )
2120adantr 452 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR+ )
224, 21rerpdivcld 10631 . . . . . 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 21245 . . . . . . . . 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 9971 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  e.  RR )
33 elfzuz 11011 . . . . . . . 8  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
34 nnuz 10477 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
3534uztrn2 10459 . . . . . . . 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 9971 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  RR )
38 flltp1 11164 . . . . . . . 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 4212 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  M
)
41 elfzle1 11016 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  M  <_  J )
4241adantl 453 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  <_  J )
4322, 32, 37, 40, 42ltletrd 9186 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  J
)
444, 37, 21ltdivmul2d 10652 . . . . 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 11015 . . . . . 6  |-  ( J  e.  ( M ... N )  ->  J  e.  ZZ )
4847adantl 453 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  ZZ )
49 relogexp 20443 . . . . 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 4198 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  X )  <  ( log `  ( K ^ J ) ) )
5246, 48rpexpcld 11501 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( K ^ J )  e.  RR+ )
53 logltb 20447 . . . 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 6056 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  ( log `  ( K ^ J ) ) )  =  ( 2  x.  ( J  x.  ( log `  K ) ) ) )
57 2z 10268 . . . . . . . 8  |-  2  e.  ZZ
58 relogexp 20443 . . . . . . . 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 10026 . . . . . . . . 9  |-  2  e.  CC
6160a1i 11 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  CC )
6237recnd 9070 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  CC )
6346relogcld 20471 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR )
6463recnd 9070 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  CC )
6561, 62, 64mulassd 9067 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  x.  ( log `  K
) )  =  ( 2  x.  ( J  x.  ( log `  K
) ) ) )
6656, 59, 653eqtr4d 2446 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  =  ( ( 2  x.  J
)  x.  ( log `  K ) ) )
67 elfzle2 11017 . . . . . . . . . . 11  |-  ( J  e.  ( M ... N )  ->  J  <_  N )
6867adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  N )
6968, 28syl6breq 4211 . . . . . . . . 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 21244 . . . . . . . . . . . . . . 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 20471 . . . . . . . . . . . 12  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  Z )  e.  RR )
7473, 21rerpdivcld 10631 . . . . . . . . . . 11  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  Z )  / 
( log `  K
) )  e.  RR )
7574rehalfcld 10170 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( log `  Z
)  /  ( log `  K ) )  / 
2 )  e.  RR )
76 flge 11169 . . . . . . . . . 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 10025 . . . . . . . . . 10  |-  2  e.  RR
8079a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  RR )
81 2pos 10038 . . . . . . . . . 10  |-  0  <  2
8281a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  0  <  2 )
83 lemuldiv2 9846 . . . . . . . . 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 9031 . . . . . . . . 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 10649 . . . . . . 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 4192 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  <_  ( log `  Z ) )
91 rpexpcl 11355 . . . . . . 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 20475 . . . . 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 10611 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( Z  e.  RR  /\  0  <_  Z ) )
96 resqrth 12016 . . . . 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 4198 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  <_  (
( sqr `  Z
) ^ 2 ) )
9952rprege0d 10611 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J )  e.  RR  /\  0  <_ 
( K ^ J
) ) )
10072rpsqrcld 12169 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( sqr `  Z )  e.  RR+ )
101100rprege0d 10611 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( sqr `  Z )  e.  RR  /\  0  <_ 
( sqr `  Z
) ) )
102 le2sq 11411 . . . 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 1649    e. wcel 1721   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    +oocpnf 9073    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   3c3 10006   4c4 10007   ZZcz 10238  ;cdc 10338   ZZ>=cuz 10444   RR+crp 10568   (,)cioo 10872   [,)cico 10874   ...cfz 10999   |_cfl 11156   ^cexp 11337   sqrcsqr 11993   expce 12619   _eceu 12620   logclog 20405  ψcchp 20828
This theorem is referenced by:  pntlemr  21249  pntlemj  21250  pntlemi  21251  pntlemf  21252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407
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