Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stirlinglem14 Structured version   Visualization version   Unicode version

Theorem stirlinglem14 38061
Description: The sequence  A converges to a positive real. This proves that the Stirling's formula converges to the factorial, up to a constant. In another theorem, using Wallis' formula for π& , such constant is exactly determined, thus proving the Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
stirlinglem14.1  |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
stirlinglem14.2  |-  B  =  ( n  e.  NN  |->  ( log `  ( A `
 n ) ) )
Assertion
Ref Expression
stirlinglem14  |-  E. c  e.  RR+  A  ~~>  c
Distinct variable group:    A, c
Allowed substitution hints:    A( n)    B( n, c)

Proof of Theorem stirlinglem14
Dummy variables  d 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stirlinglem14.1 . . 3  |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
2 stirlinglem14.2 . . 3  |-  B  =  ( n  e.  NN  |->  ( log `  ( A `
 n ) ) )
31, 2stirlinglem13 38060 . 2  |-  E. d  e.  RR  B  ~~>  d
4 simpl 464 . . . . 5  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  d  e.  RR )
54rpefcld 14236 . . . 4  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  ( exp `  d )  e.  RR+ )
6 nnuz 11218 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
7 1zzd 10992 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  1  e.  ZZ )
8 efcn 23477 . . . . . . 7  |-  exp  e.  ( CC -cn-> CC )
98a1i 11 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  exp  e.  ( CC -cn-> CC ) )
10 nnnn0 10900 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  n  e.  NN0 )
11 faccl 12507 . . . . . . . . . . . . 13  |-  ( n  e.  NN0  ->  ( ! `
 n )  e.  NN )
12 nncn 10639 . . . . . . . . . . . . 13  |-  ( ( ! `  n )  e.  NN  ->  ( ! `  n )  e.  CC )
1310, 11, 123syl 18 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( ! `  n )  e.  CC )
14 2cnd 10704 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  2  e.  CC )
15 nncn 10639 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  e.  CC )
1614, 15mulcld 9681 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  CC )
1716sqrtcld 13576 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( sqr `  ( 2  x.  n ) )  e.  CC )
18 epr 14337 . . . . . . . . . . . . . . . . 17  |-  _e  e.  RR+
19 rpcn 11333 . . . . . . . . . . . . . . . . 17  |-  ( _e  e.  RR+  ->  _e  e.  CC )
2018, 19ax-mp 5 . . . . . . . . . . . . . . . 16  |-  _e  e.  CC
2120a1i 11 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  _e  e.  CC )
22 0re 9661 . . . . . . . . . . . . . . . . 17  |-  0  e.  RR
23 epos 14336 . . . . . . . . . . . . . . . . 17  |-  0  <  _e
2422, 23gtneii 9764 . . . . . . . . . . . . . . . 16  |-  _e  =/=  0
2524a1i 11 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  _e  =/=  0 )
2615, 21, 25divcld 10405 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
n  /  _e )  e.  CC )
2726, 10expcld 12454 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  (
( n  /  _e ) ^ n )  e.  CC )
2817, 27mulcld 9681 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  e.  CC )
29 2rp 11330 . . . . . . . . . . . . . . . . 17  |-  2  e.  RR+
3029a1i 11 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  2  e.  RR+ )
31 nnrp 11334 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  n  e.  RR+ )
3230, 31rpmulcld 11380 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  RR+ )
3332sqrtgt0d 13551 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  0  <  ( sqr `  (
2  x.  n ) ) )
3433gt0ne0d 10199 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( sqr `  ( 2  x.  n ) )  =/=  0 )
35 nnne0 10664 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  =/=  0 )
3615, 21, 35, 25divne0d 10421 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
n  /  _e )  =/=  0 )
37 nnz 10983 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  n  e.  ZZ )
3826, 36, 37expne0d 12460 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  (
( n  /  _e ) ^ n )  =/=  0 )
3917, 27, 34, 38mulne0d 10286 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =/=  0 )
4013, 28, 39divcld 10405 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  e.  CC )
411fvmpt2 5972 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( ( ! `  n )  /  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) )  e.  CC )  ->  ( A `  n )  =  ( ( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) ) )
4240, 41mpdan 681 . . . . . . . . . 10  |-  ( n  e.  NN  ->  ( A `  n )  =  ( ( ! `
 n )  / 
( ( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
4342, 40eqeltrd 2549 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( A `  n )  e.  CC )
44 nnne0 10664 . . . . . . . . . . . 12  |-  ( ( ! `  n )  e.  NN  ->  ( ! `  n )  =/=  0 )
4510, 11, 443syl 18 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( ! `  n )  =/=  0 )
4613, 28, 45, 39divne0d 10421 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  =/=  0 )
4742, 46eqnetrd 2710 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( A `  n )  =/=  0 )
4843, 47logcld 23599 . . . . . . . 8  |-  ( n  e.  NN  ->  ( log `  ( A `  n ) )  e.  CC )
492, 48fmpti 6060 . . . . . . 7  |-  B : NN
--> CC
5049a1i 11 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  B : NN --> CC )
51 simpr 468 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  B  ~~>  d )
524recnd 9687 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  d  e.  CC )
536, 7, 9, 50, 51, 52climcncf 22010 . . . . 5  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  ( exp  o.  B )  ~~>  ( exp `  d ) )
548elexi 3041 . . . . . . . . 9  |-  exp  e.  _V
55 nnex 10637 . . . . . . . . . . 11  |-  NN  e.  _V
5655mptex 6152 . . . . . . . . . 10  |-  ( n  e.  NN  |->  ( log `  ( A `  n
) ) )  e. 
_V
572, 56eqeltri 2545 . . . . . . . . 9  |-  B  e. 
_V
5854, 57coex 6764 . . . . . . . 8  |-  ( exp 
o.  B )  e. 
_V
5958a1i 11 . . . . . . 7  |-  ( T. 
->  ( exp  o.  B
)  e.  _V )
6055mptex 6152 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( ( sqr `  ( 2  x.  n
) )  x.  (
( n  /  _e ) ^ n ) ) ) )  e.  _V
611, 60eqeltri 2545 . . . . . . . 8  |-  A  e. 
_V
6261a1i 11 . . . . . . 7  |-  ( T. 
->  A  e.  _V )
63 1zzd 10992 . . . . . . 7  |-  ( T. 
->  1  e.  ZZ )
642funmpt2 5626 . . . . . . . . . 10  |-  Fun  B
65 id 22 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  NN )
66 rabid2 2954 . . . . . . . . . . . . 13  |-  ( NN  =  { n  e.  NN  |  ( log `  ( A `  n
) )  e.  _V } 
<-> 
A. n  e.  NN  ( log `  ( A `
 n ) )  e.  _V )
671stirlinglem2 38049 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  ( A `  n )  e.  RR+ )
68 relogcl 23604 . . . . . . . . . . . . . 14  |-  ( ( A `  n )  e.  RR+  ->  ( log `  ( A `  n
) )  e.  RR )
69 elex 3040 . . . . . . . . . . . . . 14  |-  ( ( log `  ( A `
 n ) )  e.  RR  ->  ( log `  ( A `  n ) )  e. 
_V )
7067, 68, 693syl 18 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( log `  ( A `  n ) )  e. 
_V )
7166, 70mprgbir 2771 . . . . . . . . . . . 12  |-  NN  =  { n  e.  NN  |  ( log `  ( A `  n )
)  e.  _V }
722dmmpt 5337 . . . . . . . . . . . 12  |-  dom  B  =  { n  e.  NN  |  ( log `  ( A `  n )
)  e.  _V }
7371, 72eqtr4i 2496 . . . . . . . . . . 11  |-  NN  =  dom  B
7465, 73syl6eleq 2559 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  dom  B )
75 fvco 5956 . . . . . . . . . 10  |-  ( ( Fun  B  /\  k  e.  dom  B )  -> 
( ( exp  o.  B ) `  k
)  =  ( exp `  ( B `  k
) ) )
7664, 74, 75sylancr 676 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
( exp  o.  B
) `  k )  =  ( exp `  ( B `  k )
) )
771a1i 11 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  A  =  ( n  e.  NN  |->  ( ( ! `
 n )  / 
( ( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) ) )
78 simpr 468 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN  /\  n  =  k )  ->  n  =  k )
7978fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ! `  n
)  =  ( ! `
 k ) )
8078oveq2d 6324 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( 2  x.  n
)  =  ( 2  x.  k ) )
8180fveq2d 5883 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( sqr `  (
2  x.  n ) )  =  ( sqr `  ( 2  x.  k
) ) )
8278oveq1d 6323 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( n  /  _e )  =  ( k  /  _e ) )
8382, 78oveq12d 6326 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ( n  /  _e ) ^ n )  =  ( ( k  /  _e ) ^
k ) )
8481, 83oveq12d 6326 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) )
8579, 84oveq12d 6326 . . . . . . . . . . . . . 14  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ( ! `  n )  /  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) )  =  ( ( ! `  k )  /  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) ) )
86 nnnn0 10900 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  k  e.  NN0 )
87 faccl 12507 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
88 nncn 10639 . . . . . . . . . . . . . . . 16  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
8986, 87, 883syl 18 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  ( ! `  k )  e.  CC )
90 2cnd 10704 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  2  e.  CC )
91 nncn 10639 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  e.  CC )
9290, 91mulcld 9681 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
2  x.  k )  e.  CC )
9392sqrtcld 13576 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  ( sqr `  ( 2  x.  k ) )  e.  CC )
9420a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  _e  e.  CC )
9524a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  _e  =/=  0 )
9691, 94, 95divcld 10405 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
k  /  _e )  e.  CC )
9796, 86expcld 12454 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
( k  /  _e ) ^ k )  e.  CC )
9893, 97mulcld 9681 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) )  e.  CC )
9929a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  ->  2  e.  RR+ )
100 nnrp 11334 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  ->  k  e.  RR+ )
10199, 100rpmulcld 11380 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  (
2  x.  k )  e.  RR+ )
102101sqrtgt0d 13551 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  0  <  ( sqr `  (
2  x.  k ) ) )
103102gt0ne0d 10199 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  ( sqr `  ( 2  x.  k ) )  =/=  0 )
104 nnne0 10664 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  =/=  0 )
10591, 94, 104, 95divne0d 10421 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
k  /  _e )  =/=  0 )
106 nnz 10983 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  k  e.  ZZ )
10796, 105, 106expne0d 12460 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
( k  /  _e ) ^ k )  =/=  0 )
10893, 97, 103, 107mulne0d 10286 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) )  =/=  0 )
10989, 98, 108divcld 10405 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  (
( ! `  k
)  /  ( ( sqr `  ( 2  x.  k ) )  x.  ( ( k  /  _e ) ^
k ) ) )  e.  CC )
11077, 85, 65, 109fvmptd 5969 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  ( A `  k )  =  ( ( ! `
 k )  / 
( ( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) ) ) )
111110, 109eqeltrd 2549 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( A `  k )  e.  CC )
112 nnne0 10664 . . . . . . . . . . . . . . 15  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  =/=  0 )
11386, 87, 1123syl 18 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( ! `  k )  =/=  0 )
11489, 98, 113, 108divne0d 10421 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
( ! `  k
)  /  ( ( sqr `  ( 2  x.  k ) )  x.  ( ( k  /  _e ) ^
k ) ) )  =/=  0 )
115110, 114eqnetrd 2710 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( A `  k )  =/=  0 )
116111, 115logcld 23599 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  ( log `  ( A `  k ) )  e.  CC )
117 nfcv 2612 . . . . . . . . . . . 12  |-  F/_ n
k
118 nfcv 2612 . . . . . . . . . . . . 13  |-  F/_ n log
119 nfmpt1 4485 . . . . . . . . . . . . . . 15  |-  F/_ n
( n  e.  NN  |->  ( ( ! `  n )  /  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
1201, 119nfcxfr 2610 . . . . . . . . . . . . . 14  |-  F/_ n A
121120, 117nffv 5886 . . . . . . . . . . . . 13  |-  F/_ n
( A `  k
)
122118, 121nffv 5886 . . . . . . . . . . . 12  |-  F/_ n
( log `  ( A `  k )
)
123 fveq2 5879 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  ( A `  n )  =  ( A `  k ) )
124123fveq2d 5883 . . . . . . . . . . . 12  |-  ( n  =  k  ->  ( log `  ( A `  n ) )  =  ( log `  ( A `  k )
) )
125117, 122, 124, 2fvmptf 5981 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  ( log `  ( A `
 k ) )  e.  CC )  -> 
( B `  k
)  =  ( log `  ( A `  k
) ) )
126116, 125mpdan 681 . . . . . . . . . 10  |-  ( k  e.  NN  ->  ( B `  k )  =  ( log `  ( A `  k )
) )
127126fveq2d 5883 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( exp `  ( B `  k ) )  =  ( exp `  ( log `  ( A `  k ) ) ) )
128 eflog 23605 . . . . . . . . . 10  |-  ( ( ( A `  k
)  e.  CC  /\  ( A `  k )  =/=  0 )  -> 
( exp `  ( log `  ( A `  k ) ) )  =  ( A `  k ) )
129111, 115, 128syl2anc 673 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( exp `  ( log `  ( A `  k )
) )  =  ( A `  k ) )
13076, 127, 1293eqtrd 2509 . . . . . . . 8  |-  ( k  e.  NN  ->  (
( exp  o.  B
) `  k )  =  ( A `  k ) )
131130adantl 473 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( exp  o.  B
) `  k )  =  ( A `  k ) )
1326, 59, 62, 63, 131climeq 13708 . . . . . 6  |-  ( T. 
->  ( ( exp  o.  B )  ~~>  ( exp `  d )  <->  A  ~~>  ( exp `  d ) ) )
133132trud 1461 . . . . 5  |-  ( ( exp  o.  B )  ~~>  ( exp `  d
)  <->  A  ~~>  ( exp `  d ) )
13453, 133sylib 201 . . . 4  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  A  ~~>  ( exp `  d ) )
135 breq2 4399 . . . . 5  |-  ( c  =  ( exp `  d
)  ->  ( A  ~~>  c 
<->  A  ~~>  ( exp `  d
) ) )
136135rspcev 3136 . . . 4  |-  ( ( ( exp `  d
)  e.  RR+  /\  A  ~~>  ( exp `  d ) )  ->  E. c  e.  RR+  A  ~~>  c )
1375, 134, 136syl2anc 673 . . 3  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  E. c  e.  RR+  A  ~~>  c )
138137rexlimiva 2868 . 2  |-  ( E. d  e.  RR  B  ~~>  d  ->  E. c  e.  RR+  A  ~~>  c )
1393, 138ax-mp 5 1  |-  E. c  e.  RR+  A  ~~>  c
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   T. wtru 1453    e. wcel 1904    =/= wne 2641   E.wrex 2757   {crab 2760   _Vcvv 3031   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839    o. ccom 4843   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   RR+crp 11325   ^cexp 12310   !cfa 12497   sqrcsqrt 13373    ~~> cli 13625   expce 14191   _eceu 14192   -cn->ccncf 21986   logclog 23583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-e 14199  df-sin 14200  df-cos 14201  df-tan 14202  df-pi 14203  df-dvds 14383  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-ulm 23411  df-log 23585  df-cxp 23586
This theorem is referenced by:  stirling  38063
  Copyright terms: Public domain W3C validator