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Theorem stirlinglem14 31758
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 31757 . 2  |-  E. d  e.  RR  B  ~~>  d
4 simpl 457 . . . . 5  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  d  e.  RR )
54rpefcld 13717 . . . 4  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  ( exp `  d )  e.  RR+ )
6 nnuz 11125 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
7 1zzd 10901 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  1  e.  ZZ )
8 efcn 22710 . . . . . . 7  |-  exp  e.  ( CC -cn-> CC )
98a1i 11 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  exp  e.  ( CC -cn-> CC ) )
10 nnnn0 10808 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  n  e.  NN0 )
11 faccl 12342 . . . . . . . . . . . . 13  |-  ( n  e.  NN0  ->  ( ! `
 n )  e.  NN )
12 nncn 10550 . . . . . . . . . . . . 13  |-  ( ( ! `  n )  e.  NN  ->  ( ! `  n )  e.  CC )
1310, 11, 123syl 20 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( ! `  n )  e.  CC )
14 2cnd 10614 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  2  e.  CC )
15 nncn 10550 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  e.  CC )
1614, 15mulcld 9619 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  CC )
1716sqrtcld 13247 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( sqr `  ( 2  x.  n ) )  e.  CC )
18 epr 13818 . . . . . . . . . . . . . . . . 17  |-  _e  e.  RR+
19 rpcn 11237 . . . . . . . . . . . . . . . . 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 9599 . . . . . . . . . . . . . . . . 17  |-  0  e.  RR
23 epos 13817 . . . . . . . . . . . . . . . . 17  |-  0  <  _e
2422, 23gtneii 9699 . . . . . . . . . . . . . . . 16  |-  _e  =/=  0
2524a1i 11 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  _e  =/=  0 )
2615, 21, 25divcld 10326 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
n  /  _e )  e.  CC )
2726, 10expcld 12289 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  (
( n  /  _e ) ^ n )  e.  CC )
2817, 27mulcld 9619 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  e.  CC )
29 2rp 11234 . . . . . . . . . . . . . . . . 17  |-  2  e.  RR+
3029a1i 11 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  2  e.  RR+ )
31 nnrp 11238 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  n  e.  RR+ )
3230, 31rpmulcld 11281 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  RR+ )
3332sqrtgt0d 13223 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  0  <  ( sqr `  (
2  x.  n ) ) )
3433gt0ne0d 10123 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( sqr `  ( 2  x.  n ) )  =/=  0 )
35 nnne0 10574 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  =/=  0 )
3615, 21, 35, 25divne0d 10342 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
n  /  _e )  =/=  0 )
37 nnz 10892 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  n  e.  ZZ )
3826, 36, 37expne0d 12295 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  (
( n  /  _e ) ^ n )  =/=  0 )
3917, 27, 34, 38mulne0d 10207 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =/=  0 )
4013, 28, 39divcld 10326 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  e.  CC )
411fvmpt2 5948 . . . . . . . . . . 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 668 . . . . . . . . . 10  |-  ( n  e.  NN  ->  ( A `  n )  =  ( ( ! `
 n )  / 
( ( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
4342, 40eqeltrd 2531 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( A `  n )  e.  CC )
44 nnne0 10574 . . . . . . . . . . . 12  |-  ( ( ! `  n )  e.  NN  ->  ( ! `  n )  =/=  0 )
4510, 11, 443syl 20 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( ! `  n )  =/=  0 )
4613, 28, 45, 39divne0d 10342 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  =/=  0 )
4742, 46eqnetrd 2736 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( A `  n )  =/=  0 )
4843, 47logcld 22830 . . . . . . . 8  |-  ( n  e.  NN  ->  ( log `  ( A `  n ) )  e.  CC )
492, 48fmpti 6039 . . . . . . 7  |-  B : NN
--> CC
5049a1i 11 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  B : NN --> CC )
51 simpr 461 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  B  ~~>  d )
524recnd 9625 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  d  e.  CC )
536, 7, 9, 50, 51, 52climcncf 21277 . . . . 5  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  ( exp  o.  B )  ~~>  ( exp `  d ) )
548elexi 3105 . . . . . . . . 9  |-  exp  e.  _V
55 nnex 10548 . . . . . . . . . . 11  |-  NN  e.  _V
5655mptex 6128 . . . . . . . . . 10  |-  ( n  e.  NN  |->  ( log `  ( A `  n
) ) )  e. 
_V
572, 56eqeltri 2527 . . . . . . . . 9  |-  B  e. 
_V
5854, 57coex 6737 . . . . . . . 8  |-  ( exp 
o.  B )  e. 
_V
5958a1i 11 . . . . . . 7  |-  ( T. 
->  ( exp  o.  B
)  e.  _V )
6055mptex 6128 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( ( sqr `  ( 2  x.  n
) )  x.  (
( n  /  _e ) ^ n ) ) ) )  e.  _V
611, 60eqeltri 2527 . . . . . . . 8  |-  A  e. 
_V
6261a1i 11 . . . . . . 7  |-  ( T. 
->  A  e.  _V )
63 1zzd 10901 . . . . . . 7  |-  ( T. 
->  1  e.  ZZ )
642funmpt2 5615 . . . . . . . . . 10  |-  Fun  B
65 id 22 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  NN )
66 rabid2 3021 . . . . . . . . . . . . 13  |-  ( NN  =  { n  e.  NN  |  ( log `  ( A `  n
) )  e.  _V } 
<-> 
A. n  e.  NN  ( log `  ( A `
 n ) )  e.  _V )
671stirlinglem2 31746 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  ( A `  n )  e.  RR+ )
68 relogcl 22835 . . . . . . . . . . . . . 14  |-  ( ( A `  n )  e.  RR+  ->  ( log `  ( A `  n
) )  e.  RR )
69 elex 3104 . . . . . . . . . . . . . 14  |-  ( ( log `  ( A `
 n ) )  e.  RR  ->  ( log `  ( A `  n ) )  e. 
_V )
7067, 68, 693syl 20 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( log `  ( A `  n ) )  e. 
_V )
7166, 70mprgbir 2807 . . . . . . . . . . . 12  |-  NN  =  { n  e.  NN  |  ( log `  ( A `  n )
)  e.  _V }
722dmmpt 5492 . . . . . . . . . . . 12  |-  dom  B  =  { n  e.  NN  |  ( log `  ( A `  n )
)  e.  _V }
7371, 72eqtr4i 2475 . . . . . . . . . . 11  |-  NN  =  dom  B
7465, 73syl6eleq 2541 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  dom  B )
75 fvco 5934 . . . . . . . . . 10  |-  ( ( Fun  B  /\  k  e.  dom  B )  -> 
( ( exp  o.  B ) `  k
)  =  ( exp `  ( B `  k
) ) )
7664, 74, 75sylancr 663 . . . . . . . . 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 461 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN  /\  n  =  k )  ->  n  =  k )
7978fveq2d 5860 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ! `  n
)  =  ( ! `
 k ) )
8078oveq2d 6297 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( 2  x.  n
)  =  ( 2  x.  k ) )
8180fveq2d 5860 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( sqr `  (
2  x.  n ) )  =  ( sqr `  ( 2  x.  k
) ) )
8278oveq1d 6296 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( n  /  _e )  =  ( k  /  _e ) )
8382, 78oveq12d 6299 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ( n  /  _e ) ^ n )  =  ( ( k  /  _e ) ^
k ) )
8481, 83oveq12d 6299 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) )
8579, 84oveq12d 6299 . . . . . . . . . . . . . 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 10808 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  k  e.  NN0 )
87 faccl 12342 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
88 nncn 10550 . . . . . . . . . . . . . . . 16  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
8986, 87, 883syl 20 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  ( ! `  k )  e.  CC )
90 2cnd 10614 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  2  e.  CC )
91 nncn 10550 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  e.  CC )
9290, 91mulcld 9619 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
2  x.  k )  e.  CC )
9392sqrtcld 13247 . . . . . . . . . . . . . . . 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 10326 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
k  /  _e )  e.  CC )
9796, 86expcld 12289 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
( k  /  _e ) ^ k )  e.  CC )
9893, 97mulcld 9619 . . . . . . . . . . . . . . 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 11238 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  ->  k  e.  RR+ )
10199, 100rpmulcld 11281 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  (
2  x.  k )  e.  RR+ )
102101sqrtgt0d 13223 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  0  <  ( sqr `  (
2  x.  k ) ) )
103102gt0ne0d 10123 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  ( sqr `  ( 2  x.  k ) )  =/=  0 )
104 nnne0 10574 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  =/=  0 )
10591, 94, 104, 95divne0d 10342 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
k  /  _e )  =/=  0 )
106 nnz 10892 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  k  e.  ZZ )
10796, 105, 106expne0d 12295 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
( k  /  _e ) ^ k )  =/=  0 )
10893, 97, 103, 107mulne0d 10207 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) )  =/=  0 )
10989, 98, 108divcld 10326 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  (
( ! `  k
)  /  ( ( sqr `  ( 2  x.  k ) )  x.  ( ( k  /  _e ) ^
k ) ) )  e.  CC )
11077, 85, 65, 109fvmptd 5946 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  ( A `  k )  =  ( ( ! `
 k )  / 
( ( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) ) ) )
111110, 109eqeltrd 2531 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( A `  k )  e.  CC )
112 nnne0 10574 . . . . . . . . . . . . . . 15  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  =/=  0 )
11386, 87, 1123syl 20 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( ! `  k )  =/=  0 )
11489, 98, 113, 108divne0d 10342 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
( ! `  k
)  /  ( ( sqr `  ( 2  x.  k ) )  x.  ( ( k  /  _e ) ^
k ) ) )  =/=  0 )
115110, 114eqnetrd 2736 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( A `  k )  =/=  0 )
116111, 115logcld 22830 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  ( log `  ( A `  k ) )  e.  CC )
117 nfcv 2605 . . . . . . . . . . . 12  |-  F/_ n
k
118 nfcv 2605 . . . . . . . . . . . . 13  |-  F/_ n log
119 nfmpt1 4526 . . . . . . . . . . . . . . 15  |-  F/_ n
( n  e.  NN  |->  ( ( ! `  n )  /  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
1201, 119nfcxfr 2603 . . . . . . . . . . . . . 14  |-  F/_ n A
121120, 117nffv 5863 . . . . . . . . . . . . 13  |-  F/_ n
( A `  k
)
122118, 121nffv 5863 . . . . . . . . . . . 12  |-  F/_ n
( log `  ( A `  k )
)
123 fveq2 5856 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  ( A `  n )  =  ( A `  k ) )
124123fveq2d 5860 . . . . . . . . . . . 12  |-  ( n  =  k  ->  ( log `  ( A `  n ) )  =  ( log `  ( A `  k )
) )
125117, 122, 124, 2fvmptf 5957 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  ( log `  ( A `
 k ) )  e.  CC )  -> 
( B `  k
)  =  ( log `  ( A `  k
) ) )
126116, 125mpdan 668 . . . . . . . . . 10  |-  ( k  e.  NN  ->  ( B `  k )  =  ( log `  ( A `  k )
) )
127126fveq2d 5860 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( exp `  ( B `  k ) )  =  ( exp `  ( log `  ( A `  k ) ) ) )
128 eflog 22836 . . . . . . . . . 10  |-  ( ( ( A `  k
)  e.  CC  /\  ( A `  k )  =/=  0 )  -> 
( exp `  ( log `  ( A `  k ) ) )  =  ( A `  k ) )
129111, 115, 128syl2anc 661 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( exp `  ( log `  ( A `  k )
) )  =  ( A `  k ) )
13076, 127, 1293eqtrd 2488 . . . . . . . 8  |-  ( k  e.  NN  ->  (
( exp  o.  B
) `  k )  =  ( A `  k ) )
131130adantl 466 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( exp  o.  B
) `  k )  =  ( A `  k ) )
1326, 59, 62, 63, 131climeq 13369 . . . . . 6  |-  ( T. 
->  ( ( exp  o.  B )  ~~>  ( exp `  d )  <->  A  ~~>  ( exp `  d ) ) )
133132trud 1392 . . . . 5  |-  ( ( exp  o.  B )  ~~>  ( exp `  d
)  <->  A  ~~>  ( exp `  d ) )
13453, 133sylib 196 . . . 4  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  A  ~~>  ( exp `  d ) )
135 breq2 4441 . . . . 5  |-  ( c  =  ( exp `  d
)  ->  ( A  ~~>  c 
<->  A  ~~>  ( exp `  d
) ) )
136135rspcev 3196 . . . 4  |-  ( ( ( exp `  d
)  e.  RR+  /\  A  ~~>  ( exp `  d ) )  ->  E. c  e.  RR+  A  ~~>  c )
1375, 134, 136syl2anc 661 . . 3  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  E. c  e.  RR+  A  ~~>  c )
138137rexlimiva 2931 . 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 184    /\ wa 369    = wceq 1383   T. wtru 1384    e. wcel 1804    =/= wne 2638   E.wrex 2794   {crab 2797   _Vcvv 3095   class class class wbr 4437    |-> cmpt 4495   dom cdm 4989    o. ccom 4993   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    x. cmul 9500    / cdiv 10212   NNcn 10542   2c2 10591   NN0cn0 10801   RR+crp 11229   ^cexp 12145   !cfa 12332   sqrcsqrt 13045    ~~> cli 13286   expce 13675   _eceu 13676   -cn->ccncf 21253   logclog 22814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ioc 11543  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-e 13682  df-sin 13683  df-cos 13684  df-tan 13685  df-pi 13686  df-dvds 13864  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-haus 19689  df-cmp 19760  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-xms 20696  df-ms 20697  df-tms 20698  df-cncf 21255  df-limc 22143  df-dv 22144  df-ulm 22644  df-log 22816  df-cxp 22817
This theorem is referenced by:  stirling  31760
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