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.

Step	Hyp	Ref	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 ) ) )
3	1, 2	stirlinglem13 31757	. 2 |- E. d e. RR B ~~> d
4		simpl 457	. . . . 5 |- ( ( d e. RR /\ B ~~> d ) -> d e. RR )
5	4	rpefcld 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 )
9	8	a1i 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 )
13	10, 11, 12	3syl 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 )
16	14, 15	mulcld 9619	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( 2 x. n ) e. CC )
17	16	sqrtcld 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 )
20	18, 19	ax-mp 5	. . . . . . . . . . . . . . . 16 |- _e e. CC
21	20	a1i 11	. . . . . . . . . . . . . . 15 |- ( n e. NN -> _e e. CC )
22		0re 9599	. . . . . . . . . . . . . . . . 17 |- 0 e. RR
23		epos 13817	. . . . . . . . . . . . . . . . 17 |- 0 < _e
24	22, 23	gtneii 9699	. . . . . . . . . . . . . . . 16 |- _e =/= 0
25	24	a1i 11	. . . . . . . . . . . . . . 15 |- ( n e. NN -> _e =/= 0 )
26	15, 21, 25	divcld 10326	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( n / _e ) e. CC )
27	26, 10	expcld 12289	. . . . . . . . . . . . 13 |- ( n e. NN -> ( ( n / _e ) ^ n ) e. CC )
28	17, 27	mulcld 9619	. . . . . . . . . . . 12 |- ( n e. NN -> ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) e. CC )
29		2rp 11234	. . . . . . . . . . . . . . . . 17 |- 2 e. RR+
30	29	a1i 11	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> 2 e. RR+ )
31		nnrp 11238	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> n e. RR+ )
32	30, 31	rpmulcld 11281	. . . . . . . . . . . . . . 15 |- ( n e. NN -> ( 2 x. n ) e. RR+ )
33	32	sqrtgt0d 13223	. . . . . . . . . . . . . 14 |- ( n e. NN -> 0 < ( sqr ` ( 2 x. n ) ) )
34	33	gt0ne0d 10123	. . . . . . . . . . . . 13 |- ( n e. NN -> ( sqr ` ( 2 x. n ) ) =/= 0 )
35		nnne0 10574	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n =/= 0 )
36	15, 21, 35, 25	divne0d 10342	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( n / _e ) =/= 0 )
37		nnz 10892	. . . . . . . . . . . . . 14 |- ( n e. NN -> n e. ZZ )
38	26, 36, 37	expne0d 12295	. . . . . . . . . . . . 13 |- ( n e. NN -> ( ( n / _e ) ^ n ) =/= 0 )
39	17, 27, 34, 38	mulne0d 10207	. . . . . . . . . . . 12 |- ( n e. NN -> ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) =/= 0 )
40	13, 28, 39	divcld 10326	. . . . . . . . . . 11 |- ( n e. NN -> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) e. CC )
41	1	fvmpt2 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 ) ) ) )
42	40, 41	mpdan 668	. . . . . . . . . 10 |- ( n e. NN -> ( A ` n ) = ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) )
43	42, 40	eqeltrd 2531	. . . . . . . . 9 |- ( n e. NN -> ( A ` n ) e. CC )
44		nnne0 10574	. . . . . . . . . . . 12 |- ( ( ! ` n ) e. NN -> ( ! ` n ) =/= 0 )
45	10, 11, 44	3syl 20	. . . . . . . . . . 11 |- ( n e. NN -> ( ! ` n ) =/= 0 )
46	13, 28, 45, 39	divne0d 10342	. . . . . . . . . 10 |- ( n e. NN -> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) =/= 0 )
47	42, 46	eqnetrd 2736	. . . . . . . . 9 |- ( n e. NN -> ( A ` n ) =/= 0 )
48	43, 47	logcld 22830	. . . . . . . 8 |- ( n e. NN -> ( log ` ( A ` n ) ) e. CC )
49	2, 48	fmpti 6039	. . . . . . 7 |- B : NN --> CC
50	49	a1i 11	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> B : NN --> CC )
51		simpr 461	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> B ~~> d )
52	4	recnd 9625	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> d e. CC )
53	6, 7, 9, 50, 51, 52	climcncf 21277	. . . . 5 |- ( ( d e. RR /\ B ~~> d ) -> ( exp o. B ) ~~> ( exp ` d ) )
54	8	elexi 3105	. . . . . . . . 9 |- exp e. _V
55		nnex 10548	. . . . . . . . . . 11 |- NN e. _V
56	55	mptex 6128	. . . . . . . . . 10 |- ( n e. NN |-> ( log ` ( A ` n ) ) ) e. _V
57	2, 56	eqeltri 2527	. . . . . . . . 9 |- B e. _V
58	54, 57	coex 6737	. . . . . . . 8 |- ( exp o. B ) e. _V
59	58	a1i 11	. . . . . . 7 |- ( T. -> ( exp o. B ) e. _V )
60	55	mptex 6128	. . . . . . . . 9 |- ( n e. NN |-> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) ) e. _V
61	1, 60	eqeltri 2527	. . . . . . . 8 |- A e. _V
62	61	a1i 11	. . . . . . 7 |- ( T. -> A e. _V )
63		1zzd 10901	. . . . . . 7 |- ( T. -> 1 e. ZZ )
64	2	funmpt2 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 )
67	1	stirlinglem2 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 )
70	67, 68, 69	3syl 20	. . . . . . . . . . . . 13 |- ( n e. NN -> ( log ` ( A ` n ) ) e. _V )
71	66, 70	mprgbir 2807	. . . . . . . . . . . 12 |- NN = { n e. NN | ( log ` ( A ` n ) ) e. _V }
72	2	dmmpt 5492	. . . . . . . . . . . 12 |- dom B = { n e. NN | ( log ` ( A ` n ) ) e. _V }
73	71, 72	eqtr4i 2475	. . . . . . . . . . 11 |- NN = dom B
74	65, 73	syl6eleq 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 ) ) )
76	64, 74, 75	sylancr 663	. . . . . . . . 9 |- ( k e. NN -> ( ( exp o. B ) ` k ) = ( exp ` ( B ` k ) ) )
77	1	a1i 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 )
79	78	fveq2d 5860	. . . . . . . . . . . . . . 15 |- ( ( k e. NN /\ n = k ) -> ( ! ` n ) = ( ! ` k ) )
80	78	oveq2d 6297	. . . . . . . . . . . . . . . . 17 |- ( ( k e. NN /\ n = k ) -> ( 2 x. n ) = ( 2 x. k ) )
81	80	fveq2d 5860	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN /\ n = k ) -> ( sqr ` ( 2 x. n ) ) = ( sqr ` ( 2 x. k ) ) )
82	78	oveq1d 6296	. . . . . . . . . . . . . . . . 17 |- ( ( k e. NN /\ n = k ) -> ( n / _e ) = ( k / _e ) )
83	82, 78	oveq12d 6299	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN /\ n = k ) -> ( ( n / _e ) ^ n ) = ( ( k / _e ) ^ k ) )
84	81, 83	oveq12d 6299	. . . . . . . . . . . . . . 15 |- ( ( k e. NN /\ n = k ) -> ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) = ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) )
85	79, 84	oveq12d 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 )
89	86, 87, 88	3syl 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 )
92	90, 91	mulcld 9619	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> ( 2 x. k ) e. CC )
93	92	sqrtcld 13247	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( sqr ` ( 2 x. k ) ) e. CC )
94	20	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> _e e. CC )
95	24	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> _e =/= 0 )
96	91, 94, 95	divcld 10326	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> ( k / _e ) e. CC )
97	96, 86	expcld 12289	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( ( k / _e ) ^ k ) e. CC )
98	93, 97	mulcld 9619	. . . . . . . . . . . . . . 15 |- ( k e. NN -> ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) e. CC )
99	29	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( k e. NN -> 2 e. RR+ )
100		nnrp 11238	. . . . . . . . . . . . . . . . . . 19 |- ( k e. NN -> k e. RR+ )
101	99, 100	rpmulcld 11281	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> ( 2 x. k ) e. RR+ )
102	101	sqrtgt0d 13223	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> 0 < ( sqr ` ( 2 x. k ) ) )
103	102	gt0ne0d 10123	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( sqr ` ( 2 x. k ) ) =/= 0 )
104		nnne0 10574	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> k =/= 0 )
105	91, 94, 104, 95	divne0d 10342	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> ( k / _e ) =/= 0 )
106		nnz 10892	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> k e. ZZ )
107	96, 105, 106	expne0d 12295	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( ( k / _e ) ^ k ) =/= 0 )
108	93, 97, 103, 107	mulne0d 10207	. . . . . . . . . . . . . . 15 |- ( k e. NN -> ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) =/= 0 )
109	89, 98, 108	divcld 10326	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) e. CC )
110	77, 85, 65, 109	fvmptd 5946	. . . . . . . . . . . . 13 |- ( k e. NN -> ( A ` k ) = ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) )
111	110, 109	eqeltrd 2531	. . . . . . . . . . . 12 |- ( k e. NN -> ( A ` k ) e. CC )
112		nnne0 10574	. . . . . . . . . . . . . . 15 |- ( ( ! ` k ) e. NN -> ( ! ` k ) =/= 0 )
113	86, 87, 112	3syl 20	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( ! ` k ) =/= 0 )
114	89, 98, 113, 108	divne0d 10342	. . . . . . . . . . . . 13 |- ( k e. NN -> ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) =/= 0 )
115	110, 114	eqnetrd 2736	. . . . . . . . . . . 12 |- ( k e. NN -> ( A ` k ) =/= 0 )
116	111, 115	logcld 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 ) ) ) )
120	1, 119	nfcxfr 2603	. . . . . . . . . . . . . 14 |- F/_ n A
121	120, 117	nffv 5863	. . . . . . . . . . . . 13 |- F/_ n ( A ` k )
122	118, 121	nffv 5863	. . . . . . . . . . . 12 |- F/_ n ( log ` ( A ` k ) )
123		fveq2 5856	. . . . . . . . . . . . 13 |- ( n = k -> ( A ` n ) = ( A ` k ) )
124	123	fveq2d 5860	. . . . . . . . . . . 12 |- ( n = k -> ( log ` ( A ` n ) ) = ( log ` ( A ` k ) ) )
125	117, 122, 124, 2	fvmptf 5957	. . . . . . . . . . 11 |- ( ( k e. NN /\ ( log ` ( A ` k ) ) e. CC ) -> ( B ` k ) = ( log ` ( A ` k ) ) )
126	116, 125	mpdan 668	. . . . . . . . . 10 |- ( k e. NN -> ( B ` k ) = ( log ` ( A ` k ) ) )
127	126	fveq2d 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 ) )
129	111, 115, 128	syl2anc 661	. . . . . . . . 9 |- ( k e. NN -> ( exp ` ( log ` ( A ` k ) ) ) = ( A ` k ) )
130	76, 127, 129	3eqtrd 2488	. . . . . . . 8 |- ( k e. NN -> ( ( exp o. B ) ` k ) = ( A ` k ) )
131	130	adantl 466	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> ( ( exp o. B ) ` k ) = ( A ` k ) )
132	6, 59, 62, 63, 131	climeq 13369	. . . . . 6 |- ( T. -> ( ( exp o. B ) ~~> ( exp ` d ) <-> A ~~> ( exp ` d ) ) )
133	132	trud 1392	. . . . 5 |- ( ( exp o. B ) ~~> ( exp ` d ) <-> A ~~> ( exp ` d ) )
134	53, 133	sylib 196	. . . 4 |- ( ( d e. RR /\ B ~~> d ) -> A ~~> ( exp ` d ) )
135		breq2 4441	. . . . 5 |- ( c = ( exp ` d ) -> ( A ~~> c <-> A ~~> ( exp ` d ) ) )
136	135	rspcev 3196	. . . 4 |- ( ( ( exp ` d ) e. RR+ /\ A ~~> ( exp ` d ) ) -> E. c e. RR+ A ~~> c )
137	5, 134, 136	syl2anc 661	. . 3 |- ( ( d e. RR /\ B ~~> d ) -> E. c e. RR+ A ~~> c )
138	137	rexlimiva 2931	. 2 |- ( E. d e. RR B ~~> d -> E. c e. RR+ A ~~> c )
139	3, 138	ax-mp 5	1 |- E. c e. RR+ A ~~> c

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