MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pntlemh Unicode version

Theorem pntlemh 20750
Description: Lemma for pnt 20765. 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 445 . . . . . . . . 9  |-  ( ph  ->  X  e.  RR+ )
32adantr 451 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  X  e.  RR+ )
43relogcld 19976 . . . . . . 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 20746 . . . . . . . . . . 11  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1615simp2d 968 . . . . . . . . . 10  |-  ( ph  ->  K  e.  RR+ )
1716rpred 10392 . . . . . . . . 9  |-  ( ph  ->  K  e.  RR )
1815simp3d 969 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
1918simp2d 968 . . . . . . . . 9  |-  ( ph  ->  1  <  K )
2017, 19rplogcld 19982 . . . . . . . 8  |-  ( ph  ->  ( log `  K
)  e.  RR+ )
2120adantr 451 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR+ )
224, 21rerpdivcld 10419 . . . . . 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 20749 . . . . . . . . 9  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
3029simp1d 967 . . . . . . . 8  |-  ( ph  ->  M  e.  NN )
3130adantr 451 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  e.  NN )
3231nnred 9763 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  e.  RR )
33 elfzuz 10796 . . . . . . . 8  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
34 nnuz 10265 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
3534uztrn2 10247 . . . . . . . 8  |-  ( ( M  e.  NN  /\  J  e.  ( ZZ>= `  M ) )  ->  J  e.  NN )
3630, 33, 35syl2an 463 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  NN )
3736nnred 9763 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  RR )
38 flltp1 10934 . . . . . . . 8  |-  ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  ->  ( ( log `  X )  / 
( log `  K
) )  <  (
( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 ) )
3922, 38syl 15 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  (
( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 ) )
4039, 27syl6breqr 4065 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  M
)
41 elfzle1 10801 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  M  <_  J )
4241adantl 452 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  M  <_  J )
4322, 32, 37, 40, 42ltletrd 8978 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  X )  / 
( log `  K
) )  <  J
)
444, 37, 21ltdivmul2d 10440 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( log `  X
)  /  ( log `  K ) )  < 
J  <->  ( log `  X
)  <  ( J  x.  ( log `  K
) ) ) )
4543, 44mpbid 201 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  X )  <  ( J  x.  ( log `  K ) ) )
4616adantr 451 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  K  e.  RR+ )
47 elfzelz 10800 . . . . . 6  |-  ( J  e.  ( M ... N )  ->  J  e.  ZZ )
4847adantl 452 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  ZZ )
49 relogexp 19951 . . . . 5  |-  ( ( K  e.  RR+  /\  J  e.  ZZ )  ->  ( log `  ( K ^ J ) )  =  ( J  x.  ( log `  K ) ) )
5046, 48, 49syl2anc 642 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( K ^ J
) )  =  ( J  x.  ( log `  K ) ) )
5145, 50breqtrrd 4051 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  X )  <  ( log `  ( K ^ J ) ) )
5246, 48rpexpcld 11270 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( K ^ J )  e.  RR+ )
53 logltb 19955 . . . 4  |-  ( ( X  e.  RR+  /\  ( K ^ J )  e.  RR+ )  ->  ( X  <  ( K ^ J )  <->  ( log `  X )  <  ( log `  ( K ^ J ) ) ) )
543, 52, 53syl2anc 642 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  <->  ( log `  X
)  <  ( log `  ( K ^ J
) ) ) )
5551, 54mpbird 223 . 2  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  X  <  ( K ^ J ) )
5650oveq2d 5876 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  ( log `  ( K ^ J ) ) )  =  ( 2  x.  ( J  x.  ( log `  K ) ) ) )
57 2z 10056 . . . . . . . 8  |-  2  e.  ZZ
58 relogexp 19951 . . . . . . . 8  |-  ( ( ( K ^ J
)  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( ( K ^ J ) ^
2 ) )  =  ( 2  x.  ( log `  ( K ^ J ) ) ) )
5952, 57, 58sylancl 643 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  =  ( 2  x.  ( log `  ( K ^ J
) ) ) )
60 2cn 9818 . . . . . . . . 9  |-  2  e.  CC
6160a1i 10 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  CC )
6237recnd 8863 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  e.  CC )
6346relogcld 19976 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  RR )
6463recnd 8863 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  K )  e.  CC )
6561, 62, 64mulassd 8860 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  x.  ( log `  K
) )  =  ( 2  x.  ( J  x.  ( log `  K
) ) ) )
6656, 59, 653eqtr4d 2327 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  =  ( ( 2  x.  J
)  x.  ( log `  K ) ) )
67 elfzle2 10802 . . . . . . . . . . 11  |-  ( J  e.  ( M ... N )  ->  J  <_  N )
6867adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  N )
6968, 28syl6breq 4064 . . . . . . . . 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 20748 . . . . . . . . . . . . . . 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 967 . . . . . . . . . . . . . 14  |-  ( ph  ->  Z  e.  RR+ )
7271adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  Z  e.  RR+ )
7372relogcld 19976 . . . . . . . . . . . 12  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  Z )  e.  RR )
7473, 21rerpdivcld 10419 . . . . . . . . . . 11  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( log `  Z )  / 
( log `  K
) )  e.  RR )
7574rehalfcld 9960 . . . . . . . . . 10  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( log `  Z
)  /  ( log `  K ) )  / 
2 )  e.  RR )
76 flge 10939 . . . . . . . . . 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 642 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( J  <_  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <->  J  <_  ( |_ `  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) ) ) )
7869, 77mpbird 223 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  J  <_  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
79 2re 9817 . . . . . . . . . 10  |-  2  e.  RR
8079a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  2  e.  RR )
81 2pos 9830 . . . . . . . . . 10  |-  0  <  2
8281a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  0  <  2 )
83 lemuldiv2 9638 . . . . . . . . 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 1186 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  <_  ( ( log `  Z )  /  ( log `  K ) )  <-> 
J  <_  ( (
( log `  Z
)  /  ( log `  K ) )  / 
2 ) ) )
8578, 84mpbird 223 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  J )  <_ 
( ( log `  Z
)  /  ( log `  K ) ) )
86 remulcl 8824 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  J  e.  RR )  ->  ( 2  x.  J
)  e.  RR )
8779, 37, 86sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( 2  x.  J )  e.  RR )
8887, 73, 21lemuldivd 10437 . . . . . . 7  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( 2  x.  J
)  x.  ( log `  K ) )  <_ 
( log `  Z
)  <->  ( 2  x.  J )  <_  (
( log `  Z
)  /  ( log `  K ) ) ) )
8985, 88mpbird 223 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
2  x.  J )  x.  ( log `  K
) )  <_  ( log `  Z ) )
9066, 89eqbrtrd 4045 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( log `  ( ( K ^ J ) ^ 2 ) )  <_  ( log `  Z ) )
91 rpexpcl 11124 . . . . . . 7  |-  ( ( ( K ^ J
)  e.  RR+  /\  2  e.  ZZ )  ->  (
( K ^ J
) ^ 2 )  e.  RR+ )
9252, 57, 91sylancl 643 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  e.  RR+ )
9392, 72logled 19980 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( (
( K ^ J
) ^ 2 )  <_  Z  <->  ( log `  ( ( K ^ J ) ^ 2 ) )  <_  ( log `  Z ) ) )
9490, 93mpbird 223 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  <_  Z
)
9572rprege0d 10399 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( Z  e.  RR  /\  0  <_  Z ) )
96 resqrth 11743 . . . . 5  |-  ( ( Z  e.  RR  /\  0  <_  Z )  -> 
( ( sqr `  Z
) ^ 2 )  =  Z )
9795, 96syl 15 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( sqr `  Z ) ^
2 )  =  Z )
9894, 97breqtrrd 4051 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J ) ^
2 )  <_  (
( sqr `  Z
) ^ 2 ) )
9952rprege0d 10399 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J )  e.  RR  /\  0  <_ 
( K ^ J
) ) )
10072rpsqrcld 11896 . . . . 5  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( sqr `  Z )  e.  RR+ )
101100rprege0d 10399 . . . 4  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( sqr `  Z )  e.  RR  /\  0  <_ 
( sqr `  Z
) ) )
102 le2sq 11180 . . . 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 642 . . 3  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( ( K ^ J )  <_ 
( sqr `  Z
)  <->  ( ( K ^ J ) ^
2 )  <_  (
( sqr `  Z
) ^ 2 ) ) )
10498, 103mpbird 223 . 2  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( K ^ J )  <_  ( sqr `  Z ) )
10555, 104jca 518 1  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   class class class wbr 4025    e. cmpt 4079   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    +oocpnf 8866    < clt 8869    <_ cle 8870    - cmin 9039    / cdiv 9425   NNcn 9748   2c2 9797   3c3 9798   4c4 9799   ZZcz 10026  ;cdc 10126   ZZ>=cuz 10232   RR+crp 10356   (,)cioo 10658   [,)cico 10660   ...cfz 10784   |_cfl 10926   ^cexp 11106   sqrcsqr 11720   expce 12345   _eceu 12346   logclog 19914  ψcchp 20332
This theorem is referenced by:  pntlemr  20753  pntlemj  20754  pntlemi  20755  pntlemf  20756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-e 12352  df-sin 12353  df-cos 12354  df-pi 12356  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916
  Copyright terms: Public domain W3C validator