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Theorem pntlemh 21160
Description: Lemma for pnt 21175. 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 20385 . . . . . . 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 21156 . . . . . . . . . . 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 10580 . . . . . . . . 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 20391 . . . . . . . 8  |-  ( ph  ->  ( log `  K
)  e.  RR+ )
2120adantr 452 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR+ )
224, 21rerpdivcld 10607 . . . . . 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 21159 . . . . . . . . 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 9947 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  e.  RR )
33 elfzuz 10987 . . . . . . . 8  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
34 nnuz 10453 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
3534uztrn2 10435 . . . . . . . 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 9947 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  RR )
38 flltp1 11136 . . . . . . . 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 4193 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  M
)
41 elfzle1 10992 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  M  <_  J )
4241adantl 453 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  <_  J )
4322, 32, 37, 40, 42ltletrd 9162 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  J
)
444, 37, 21ltdivmul2d 10628 . . . . 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 10991 . . . . . 6  |-  ( J  e.  ( M ... N )  ->  J  e.  ZZ )
4847adantl 453 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  ZZ )
49 relogexp 20357 . . . . 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 4179 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  X )  <  ( log `  ( K ^ J ) ) )
5246, 48rpexpcld 11473 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( K ^ J )  e.  RR+ )
53 logltb 20361 . . . 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 6036 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  ( log `  ( K ^ J ) ) )  =  ( 2  x.  ( J  x.  ( log `  K ) ) ) )
57 2z 10244 . . . . . . . 8  |-  2  e.  ZZ
58 relogexp 20357 . . . . . . . 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 10002 . . . . . . . . 9  |-  2  e.  CC
6160a1i 11 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  CC )
6237recnd 9047 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  CC )
6346relogcld 20385 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR )
6463recnd 9047 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  CC )
6561, 62, 64mulassd 9044 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  x.  ( log `  K
) )  =  ( 2  x.  ( J  x.  ( log `  K
) ) ) )
6656, 59, 653eqtr4d 2429 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  =  ( ( 2  x.  J
)  x.  ( log `  K ) ) )
67 elfzle2 10993 . . . . . . . . . . 11  |-  ( J  e.  ( M ... N )  ->  J  <_  N )
6867adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  N )
6968, 28syl6breq 4192 . . . . . . . . 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 21158 . . . . . . . . . . . . . . 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 20385 . . . . . . . . . . . 12  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  Z )  e.  RR )
7473, 21rerpdivcld 10607 . . . . . . . . . . 11  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  Z )  / 
( log `  K
) )  e.  RR )
7574rehalfcld 10146 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( log `  Z
)  /  ( log `  K ) )  / 
2 )  e.  RR )
76 flge 11141 . . . . . . . . . 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 10001 . . . . . . . . . 10  |-  2  e.  RR
8079a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  RR )
81 2pos 10014 . . . . . . . . . 10  |-  0  <  2
8281a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  0  <  2 )
83 lemuldiv2 9822 . . . . . . . . 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 9008 . . . . . . . . 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 10625 . . . . . . 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 4173 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  <_  ( log `  Z ) )
91 rpexpcl 11327 . . . . . . 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 20389 . . . . 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 10587 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( Z  e.  RR  /\  0  <_  Z ) )
96 resqrth 11988 . . . . 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 4179 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  <_  (
( sqr `  Z
) ^ 2 ) )
9952rprege0d 10587 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J )  e.  RR  /\  0  <_ 
( K ^ J
) ) )
10072rpsqrcld 12141 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( sqr `  Z )  e.  RR+ )
101100rprege0d 10587 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( sqr `  Z )  e.  RR  /\  0  <_ 
( sqr `  Z
) ) )
102 le2sq 11383 . . . 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 1717   class class class wbr 4153    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    +oocpnf 9050    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   NNcn 9932   2c2 9981   3c3 9982   4c4 9983   ZZcz 10214  ;cdc 10314   ZZ>=cuz 10420   RR+crp 10544   (,)cioo 10848   [,)cico 10850   ...cfz 10975   |_cfl 11128   ^cexp 11309   sqrcsqr 11965   expce 12591   _eceu 12592   logclog 20319  ψcchp 20742
This theorem is referenced by:  pntlemr  21163  pntlemj  21164  pntlemi  21165  pntlemf  21166
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-sum 12407  df-ef 12597  df-e 12598  df-sin 12599  df-cos 12600  df-pi 12602  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-limc 19620  df-dv 19621  df-log 20321
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