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Theorem pntlemh 22733
Description: Lemma for pnt 22748. 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 456 . . . . . . . . 9  |-  ( ph  ->  X  e.  RR+ )
32adantr 462 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  X  e.  RR+ )
43relogcld 21957 . . . . . . 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 22729 . . . . . . . . . . 11  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1615simp2d 994 . . . . . . . . . 10  |-  ( ph  ->  K  e.  RR+ )
1716rpred 11015 . . . . . . . . 9  |-  ( ph  ->  K  e.  RR )
1815simp3d 995 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
1918simp2d 994 . . . . . . . . 9  |-  ( ph  ->  1  <  K )
2017, 19rplogcld 21963 . . . . . . . 8  |-  ( ph  ->  ( log `  K
)  e.  RR+ )
2120adantr 462 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR+ )
224, 21rerpdivcld 11042 . . . . . 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 22732 . . . . . . . . 9  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
3029simp1d 993 . . . . . . . 8  |-  ( ph  ->  M  e.  NN )
3130adantr 462 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  e.  NN )
3231nnred 10325 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  e.  RR )
33 elfzuz 11436 . . . . . . . 8  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
34 eluznn 10913 . . . . . . . 8  |-  ( ( M  e.  NN  /\  J  e.  ( ZZ>= `  M ) )  ->  J  e.  NN )
3530, 33, 34syl2an 474 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  NN )
3635nnred 10325 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  RR )
37 flltp1 11634 . . . . . . . 8  |-  ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  ->  ( ( log `  X )  / 
( log `  K
) )  <  (
( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 ) )
3822, 37syl 16 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  (
( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 ) )
3938, 27syl6breqr 4320 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  M
)
40 elfzle1 11441 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  M  <_  J )
4140adantl 463 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  <_  J )
4222, 32, 36, 39, 41ltletrd 9519 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  J
)
434, 36, 21ltdivmul2d 11063 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( log `  X
)  /  ( log `  K ) )  < 
J  <->  ( log `  X
)  <  ( J  x.  ( log `  K
) ) ) )
4442, 43mpbid 210 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  X )  <  ( J  x.  ( log `  K ) ) )
4516adantr 462 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  K  e.  RR+ )
46 elfzelz 11440 . . . . . 6  |-  ( J  e.  ( M ... N )  ->  J  e.  ZZ )
4746adantl 463 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  ZZ )
48 relogexp 21929 . . . . 5  |-  ( ( K  e.  RR+  /\  J  e.  ZZ )  ->  ( log `  ( K ^ J ) )  =  ( J  x.  ( log `  K ) ) )
4945, 47, 48syl2anc 654 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( K ^ J
) )  =  ( J  x.  ( log `  K ) ) )
5044, 49breqtrrd 4306 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  X )  <  ( log `  ( K ^ J ) ) )
5145, 47rpexpcld 12015 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( K ^ J )  e.  RR+ )
52 logltb 21933 . . . 4  |-  ( ( X  e.  RR+  /\  ( K ^ J )  e.  RR+ )  ->  ( X  <  ( K ^ J )  <->  ( log `  X )  <  ( log `  ( K ^ J ) ) ) )
533, 51, 52syl2anc 654 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  <->  ( log `  X
)  <  ( log `  ( K ^ J
) ) ) )
5450, 53mpbird 232 . 2  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  X  <  ( K ^ J ) )
5549oveq2d 6096 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  ( log `  ( K ^ J ) ) )  =  ( 2  x.  ( J  x.  ( log `  K ) ) ) )
56 2z 10666 . . . . . . . 8  |-  2  e.  ZZ
57 relogexp 21929 . . . . . . . 8  |-  ( ( ( K ^ J
)  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( ( K ^ J ) ^
2 ) )  =  ( 2  x.  ( log `  ( K ^ J ) ) ) )
5851, 56, 57sylancl 655 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  =  ( 2  x.  ( log `  ( K ^ J
) ) ) )
59 2cnd 10382 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  CC )
6036recnd 9400 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  CC )
6145relogcld 21957 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR )
6261recnd 9400 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  CC )
6359, 60, 62mulassd 9397 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  x.  ( log `  K
) )  =  ( 2  x.  ( J  x.  ( log `  K
) ) ) )
6455, 58, 633eqtr4d 2475 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  =  ( ( 2  x.  J
)  x.  ( log `  K ) ) )
65 elfzle2 11442 . . . . . . . . . . 11  |-  ( J  e.  ( M ... N )  ->  J  <_  N )
6665adantl 463 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  N )
6766, 28syl6breq 4319 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  ( |_ `  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) ) )
685, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23, 1, 24, 25, 26pntlemb 22731 . . . . . . . . . . . . . . 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
) ) ) ) )
6968simp1d 993 . . . . . . . . . . . . . 14  |-  ( ph  ->  Z  e.  RR+ )
7069adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  Z  e.  RR+ )
7170relogcld 21957 . . . . . . . . . . . 12  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  Z )  e.  RR )
7271, 21rerpdivcld 11042 . . . . . . . . . . 11  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  Z )  / 
( log `  K
) )  e.  RR )
7372rehalfcld 10559 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( log `  Z
)  /  ( log `  K ) )  / 
2 )  e.  RR )
74 flge 11639 . . . . . . . . . 10  |-  ( ( ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  e.  RR  /\  J  e.  ZZ )  ->  ( J  <_  ( ( ( log `  Z )  /  ( log `  K
) )  /  2
)  <->  J  <_  ( |_
`  ( ( ( log `  Z )  /  ( log `  K
) )  /  2
) ) ) )
7573, 47, 74syl2anc 654 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( J  <_  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <->  J  <_  ( |_ `  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) ) ) )
7667, 75mpbird 232 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
77 2re 10379 . . . . . . . . . 10  |-  2  e.  RR
7877a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  RR )
79 2pos 10401 . . . . . . . . . 10  |-  0  <  2
8079a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  0  <  2 )
81 lemuldiv2 10200 . . . . . . . . 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 ) ) )
8236, 72, 78, 80, 81syl112anc 1215 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  <_  ( ( log `  Z )  /  ( log `  K ) )  <-> 
J  <_  ( (
( log `  Z
)  /  ( log `  K ) )  / 
2 ) ) )
8376, 82mpbird 232 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  J )  <_ 
( ( log `  Z
)  /  ( log `  K ) ) )
84 remulcl 9355 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  J  e.  RR )  ->  ( 2  x.  J
)  e.  RR )
8577, 36, 84sylancr 656 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  J )  e.  RR )
8685, 71, 21lemuldivd 11060 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( 2  x.  J
)  x.  ( log `  K ) )  <_ 
( log `  Z
)  <->  ( 2  x.  J )  <_  (
( log `  Z
)  /  ( log `  K ) ) ) )
8783, 86mpbird 232 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  x.  ( log `  K
) )  <_  ( log `  Z ) )
8864, 87eqbrtrd 4300 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  <_  ( log `  Z ) )
89 rpexpcl 11868 . . . . . . 7  |-  ( ( ( K ^ J
)  e.  RR+  /\  2  e.  ZZ )  ->  (
( K ^ J
) ^ 2 )  e.  RR+ )
9051, 56, 89sylancl 655 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  e.  RR+ )
9190, 70logled 21961 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( K ^ J
) ^ 2 )  <_  Z  <->  ( log `  ( ( K ^ J ) ^ 2 ) )  <_  ( log `  Z ) ) )
9288, 91mpbird 232 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  <_  Z
)
9370rprege0d 11022 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( Z  e.  RR  /\  0  <_  Z ) )
94 resqrth 12729 . . . . 5  |-  ( ( Z  e.  RR  /\  0  <_  Z )  -> 
( ( sqr `  Z
) ^ 2 )  =  Z )
9593, 94syl 16 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( sqr `  Z ) ^
2 )  =  Z )
9692, 95breqtrrd 4306 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  <_  (
( sqr `  Z
) ^ 2 ) )
9751rprege0d 11022 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J )  e.  RR  /\  0  <_ 
( K ^ J
) ) )
9870rpsqrcld 12882 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( sqr `  Z )  e.  RR+ )
9998rprege0d 11022 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( sqr `  Z )  e.  RR  /\  0  <_ 
( sqr `  Z
) ) )
100 le2sq 11924 . . . 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 ) ) )
10197, 99, 100syl2anc 654 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J )  <_ 
( sqr `  Z
)  <->  ( ( K ^ J ) ^
2 )  <_  (
( sqr `  Z
) ^ 2 ) ) )
10296, 101mpbird 232 . 2  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( K ^ J )  <_  ( sqr `  Z ) )
10354, 102jca 529 1  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275   +oocpnf 9403    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   NNcn 10310   2c2 10359   3c3 10360   4c4 10361   ZZcz 10634  ;cdc 10743   ZZ>=cuz 10849   RR+crp 10979   (,)cioo 11288   [,)cico 11290   ...cfz 11424   |_cfl 11624   ^cexp 11849   sqrcsqr 12706   expce 13330   _eceu 13331   logclog 21891  ψcchp 22315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-e 13337  df-sin 13338  df-cos 13339  df-pi 13341  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893
This theorem is referenced by:  pntlemr  22736  pntlemj  22737  pntlemi  22738  pntlemf  22739
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