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

Theorem stirlinglem14 37769
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 37768 . 2  |-  E. d  e.  RR  B  ~~>  d
4 simpl 458 . . . . 5  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  d  e.  RR )
54rpefcld 14147 . . . 4  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  ( exp `  d )  e.  RR+ )
6 nnuz 11195 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
7 1zzd 10969 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  1  e.  ZZ )
8 efcn 23385 . . . . . . 7  |-  exp  e.  ( CC -cn-> CC )
98a1i 11 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  exp  e.  ( CC -cn-> CC ) )
10 nnnn0 10877 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  n  e.  NN0 )
11 faccl 12469 . . . . . . . . . . . . 13  |-  ( n  e.  NN0  ->  ( ! `
 n )  e.  NN )
12 nncn 10618 . . . . . . . . . . . . 13  |-  ( ( ! `  n )  e.  NN  ->  ( ! `  n )  e.  CC )
1310, 11, 123syl 18 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( ! `  n )  e.  CC )
14 2cnd 10683 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  2  e.  CC )
15 nncn 10618 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  e.  CC )
1614, 15mulcld 9664 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  CC )
1716sqrtcld 13487 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( sqr `  ( 2  x.  n ) )  e.  CC )
18 epr 14248 . . . . . . . . . . . . . . . . 17  |-  _e  e.  RR+
19 rpcn 11311 . . . . . . . . . . . . . . . . 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 9644 . . . . . . . . . . . . . . . . 17  |-  0  e.  RR
23 epos 14247 . . . . . . . . . . . . . . . . 17  |-  0  <  _e
2422, 23gtneii 9747 . . . . . . . . . . . . . . . 16  |-  _e  =/=  0
2524a1i 11 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  _e  =/=  0 )
2615, 21, 25divcld 10384 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
n  /  _e )  e.  CC )
2726, 10expcld 12416 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  (
( n  /  _e ) ^ n )  e.  CC )
2817, 27mulcld 9664 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  e.  CC )
29 2rp 11308 . . . . . . . . . . . . . . . . 17  |-  2  e.  RR+
3029a1i 11 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  2  e.  RR+ )
31 nnrp 11312 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  n  e.  RR+ )
3230, 31rpmulcld 11358 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
2  x.  n )  e.  RR+ )
3332sqrtgt0d 13463 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  0  <  ( sqr `  (
2  x.  n ) ) )
3433gt0ne0d 10179 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( sqr `  ( 2  x.  n ) )  =/=  0 )
35 nnne0 10643 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  =/=  0 )
3615, 21, 35, 25divne0d 10400 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
n  /  _e )  =/=  0 )
37 nnz 10960 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  n  e.  ZZ )
3826, 36, 37expne0d 12422 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  (
( n  /  _e ) ^ n )  =/=  0 )
3917, 27, 34, 38mulne0d 10265 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =/=  0 )
4013, 28, 39divcld 10384 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  e.  CC )
411fvmpt2 5970 . . . . . . . . . . 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 672 . . . . . . . . . 10  |-  ( n  e.  NN  ->  ( A `  n )  =  ( ( ! `
 n )  / 
( ( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
4342, 40eqeltrd 2510 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( A `  n )  e.  CC )
44 nnne0 10643 . . . . . . . . . . . 12  |-  ( ( ! `  n )  e.  NN  ->  ( ! `  n )  =/=  0 )
4510, 11, 443syl 18 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( ! `  n )  =/=  0 )
4613, 28, 45, 39divne0d 10400 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  =/=  0 )
4742, 46eqnetrd 2717 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( A `  n )  =/=  0 )
4843, 47logcld 23507 . . . . . . . 8  |-  ( n  e.  NN  ->  ( log `  ( A `  n ) )  e.  CC )
492, 48fmpti 6057 . . . . . . 7  |-  B : NN
--> CC
5049a1i 11 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  B : NN --> CC )
51 simpr 462 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  B  ~~>  d )
524recnd 9670 . . . . . 6  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  d  e.  CC )
536, 7, 9, 50, 51, 52climcncf 21919 . . . . 5  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  ( exp  o.  B )  ~~>  ( exp `  d ) )
548elexi 3091 . . . . . . . . 9  |-  exp  e.  _V
55 nnex 10616 . . . . . . . . . . 11  |-  NN  e.  _V
5655mptex 6148 . . . . . . . . . 10  |-  ( n  e.  NN  |->  ( log `  ( A `  n
) ) )  e. 
_V
572, 56eqeltri 2506 . . . . . . . . 9  |-  B  e. 
_V
5854, 57coex 6756 . . . . . . . 8  |-  ( exp 
o.  B )  e. 
_V
5958a1i 11 . . . . . . 7  |-  ( T. 
->  ( exp  o.  B
)  e.  _V )
6055mptex 6148 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( ( sqr `  ( 2  x.  n
) )  x.  (
( n  /  _e ) ^ n ) ) ) )  e.  _V
611, 60eqeltri 2506 . . . . . . . 8  |-  A  e. 
_V
6261a1i 11 . . . . . . 7  |-  ( T. 
->  A  e.  _V )
63 1zzd 10969 . . . . . . 7  |-  ( T. 
->  1  e.  ZZ )
642funmpt2 5635 . . . . . . . . . 10  |-  Fun  B
65 id 23 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  NN )
66 rabid2 3006 . . . . . . . . . . . . 13  |-  ( NN  =  { n  e.  NN  |  ( log `  ( A `  n
) )  e.  _V } 
<-> 
A. n  e.  NN  ( log `  ( A `
 n ) )  e.  _V )
671stirlinglem2 37757 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  ( A `  n )  e.  RR+ )
68 relogcl 23512 . . . . . . . . . . . . . 14  |-  ( ( A `  n )  e.  RR+  ->  ( log `  ( A `  n
) )  e.  RR )
69 elex 3090 . . . . . . . . . . . . . 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 2789 . . . . . . . . . . . 12  |-  NN  =  { n  e.  NN  |  ( log `  ( A `  n )
)  e.  _V }
722dmmpt 5346 . . . . . . . . . . . 12  |-  dom  B  =  { n  e.  NN  |  ( log `  ( A `  n )
)  e.  _V }
7371, 72eqtr4i 2454 . . . . . . . . . . 11  |-  NN  =  dom  B
7465, 73syl6eleq 2520 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  dom  B )
75 fvco 5954 . . . . . . . . . 10  |-  ( ( Fun  B  /\  k  e.  dom  B )  -> 
( ( exp  o.  B ) `  k
)  =  ( exp `  ( B `  k
) ) )
7664, 74, 75sylancr 667 . . . . . . . . 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 462 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN  /\  n  =  k )  ->  n  =  k )
7978fveq2d 5882 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ! `  n
)  =  ( ! `
 k ) )
8078oveq2d 6318 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( 2  x.  n
)  =  ( 2  x.  k ) )
8180fveq2d 5882 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( sqr `  (
2  x.  n ) )  =  ( sqr `  ( 2  x.  k
) ) )
8278oveq1d 6317 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( n  /  _e )  =  ( k  /  _e ) )
8382, 78oveq12d 6320 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ( n  /  _e ) ^ n )  =  ( ( k  /  _e ) ^
k ) )
8481, 83oveq12d 6320 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN  /\  n  =  k )  ->  ( ( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) )
8579, 84oveq12d 6320 . . . . . . . . . . . . . 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 10877 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  k  e.  NN0 )
87 faccl 12469 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
88 nncn 10618 . . . . . . . . . . . . . . . 16  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  e.  CC )
8986, 87, 883syl 18 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  ( ! `  k )  e.  CC )
90 2cnd 10683 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  2  e.  CC )
91 nncn 10618 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  e.  CC )
9290, 91mulcld 9664 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
2  x.  k )  e.  CC )
9392sqrtcld 13487 . . . . . . . . . . . . . . . 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 10384 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
k  /  _e )  e.  CC )
9796, 86expcld 12416 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
( k  /  _e ) ^ k )  e.  CC )
9893, 97mulcld 9664 . . . . . . . . . . . . . . 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 11312 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  ->  k  e.  RR+ )
10199, 100rpmulcld 11358 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  (
2  x.  k )  e.  RR+ )
102101sqrtgt0d 13463 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  0  <  ( sqr `  (
2  x.  k ) ) )
103102gt0ne0d 10179 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  ( sqr `  ( 2  x.  k ) )  =/=  0 )
104 nnne0 10643 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  =/=  0 )
10591, 94, 104, 95divne0d 10400 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
k  /  _e )  =/=  0 )
106 nnz 10960 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  k  e.  ZZ )
10796, 105, 106expne0d 12422 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
( k  /  _e ) ^ k )  =/=  0 )
10893, 97, 103, 107mulne0d 10265 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) )  =/=  0 )
10989, 98, 108divcld 10384 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  (
( ! `  k
)  /  ( ( sqr `  ( 2  x.  k ) )  x.  ( ( k  /  _e ) ^
k ) ) )  e.  CC )
11077, 85, 65, 109fvmptd 5967 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  ( A `  k )  =  ( ( ! `
 k )  / 
( ( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) ) ) )
111110, 109eqeltrd 2510 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( A `  k )  e.  CC )
112 nnne0 10643 . . . . . . . . . . . . . . 15  |-  ( ( ! `  k )  e.  NN  ->  ( ! `  k )  =/=  0 )
11386, 87, 1123syl 18 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( ! `  k )  =/=  0 )
11489, 98, 113, 108divne0d 10400 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
( ! `  k
)  /  ( ( sqr `  ( 2  x.  k ) )  x.  ( ( k  /  _e ) ^
k ) ) )  =/=  0 )
115110, 114eqnetrd 2717 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( A `  k )  =/=  0 )
116111, 115logcld 23507 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  ( log `  ( A `  k ) )  e.  CC )
117 nfcv 2584 . . . . . . . . . . . 12  |-  F/_ n
k
118 nfcv 2584 . . . . . . . . . . . . 13  |-  F/_ n log
119 nfmpt1 4510 . . . . . . . . . . . . . . 15  |-  F/_ n
( n  e.  NN  |->  ( ( ! `  n )  /  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
1201, 119nfcxfr 2582 . . . . . . . . . . . . . 14  |-  F/_ n A
121120, 117nffv 5885 . . . . . . . . . . . . 13  |-  F/_ n
( A `  k
)
122118, 121nffv 5885 . . . . . . . . . . . 12  |-  F/_ n
( log `  ( A `  k )
)
123 fveq2 5878 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  ( A `  n )  =  ( A `  k ) )
124123fveq2d 5882 . . . . . . . . . . . 12  |-  ( n  =  k  ->  ( log `  ( A `  n ) )  =  ( log `  ( A `  k )
) )
125117, 122, 124, 2fvmptf 5979 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  ( log `  ( A `
 k ) )  e.  CC )  -> 
( B `  k
)  =  ( log `  ( A `  k
) ) )
126116, 125mpdan 672 . . . . . . . . . 10  |-  ( k  e.  NN  ->  ( B `  k )  =  ( log `  ( A `  k )
) )
127126fveq2d 5882 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( exp `  ( B `  k ) )  =  ( exp `  ( log `  ( A `  k ) ) ) )
128 eflog 23513 . . . . . . . . . 10  |-  ( ( ( A `  k
)  e.  CC  /\  ( A `  k )  =/=  0 )  -> 
( exp `  ( log `  ( A `  k ) ) )  =  ( A `  k ) )
129111, 115, 128syl2anc 665 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( exp `  ( log `  ( A `  k )
) )  =  ( A `  k ) )
13076, 127, 1293eqtrd 2467 . . . . . . . 8  |-  ( k  e.  NN  ->  (
( exp  o.  B
) `  k )  =  ( A `  k ) )
131130adantl 467 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( exp  o.  B
) `  k )  =  ( A `  k ) )
1326, 59, 62, 63, 131climeq 13619 . . . . . 6  |-  ( T. 
->  ( ( exp  o.  B )  ~~>  ( exp `  d )  <->  A  ~~>  ( exp `  d ) ) )
133132trud 1446 . . . . 5  |-  ( ( exp  o.  B )  ~~>  ( exp `  d
)  <->  A  ~~>  ( exp `  d ) )
13453, 133sylib 199 . . . 4  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  A  ~~>  ( exp `  d ) )
135 breq2 4424 . . . . 5  |-  ( c  =  ( exp `  d
)  ->  ( A  ~~>  c 
<->  A  ~~>  ( exp `  d
) ) )
136135rspcev 3182 . . . 4  |-  ( ( ( exp `  d
)  e.  RR+  /\  A  ~~>  ( exp `  d ) )  ->  E. c  e.  RR+  A  ~~>  c )
1375, 134, 136syl2anc 665 . . 3  |-  ( ( d  e.  RR  /\  B 
~~>  d )  ->  E. c  e.  RR+  A  ~~>  c )
138137rexlimiva 2913 . 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 187    /\ wa 370    = wceq 1437   T. wtru 1438    e. wcel 1868    =/= wne 2618   E.wrex 2776   {crab 2779   _Vcvv 3081   class class class wbr 4420    |-> cmpt 4479   dom cdm 4850    o. ccom 4854   Fun wfun 5592   -->wf 5594   ` cfv 5598  (class class class)co 6302   CCcc 9538   RRcr 9539   0cc0 9540   1c1 9541    x. cmul 9545    / cdiv 10270   NNcn 10610   2c2 10660   NN0cn0 10870   RR+crp 11303   ^cexp 12272   !cfa 12459   sqrcsqrt 13285    ~~> cli 13536   expce 14102   _eceu 14103   -cn->ccncf 21895   logclog 23491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-ioc 11641  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13119  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-limsup 13514  df-clim 13540  df-rlim 13541  df-sum 13741  df-ef 14109  df-e 14110  df-sin 14111  df-cos 14112  df-tan 14113  df-pi 14114  df-dvds 14294  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-fbas 18955  df-fg 18956  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cld 20021  df-ntr 20022  df-cls 20023  df-nei 20101  df-lp 20139  df-perf 20140  df-cn 20230  df-cnp 20231  df-haus 20318  df-cmp 20389  df-tx 20564  df-hmeo 20757  df-fil 20848  df-fm 20940  df-flim 20941  df-flf 20942  df-xms 21322  df-ms 21323  df-tms 21324  df-cncf 21897  df-limc 22808  df-dv 22809  df-ulm 23319  df-log 23493  df-cxp 23494
This theorem is referenced by:  stirling  37771
  Copyright terms: Public domain W3C validator