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.

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 38060	. 2 |- E. d e. RR B ~~> d
4		simpl 464	. . . . 5 |- ( ( d e. RR /\ B ~~> d ) -> d e. RR )
5	4	rpefcld 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 )
9	8	a1i 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 )
13	10, 11, 12	3syl 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 )
16	14, 15	mulcld 9681	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( 2 x. n ) e. CC )
17	16	sqrtcld 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 )
20	18, 19	ax-mp 5	. . . . . . . . . . . . . . . 16 |- _e e. CC
21	20	a1i 11	. . . . . . . . . . . . . . 15 |- ( n e. NN -> _e e. CC )
22		0re 9661	. . . . . . . . . . . . . . . . 17 |- 0 e. RR
23		epos 14336	. . . . . . . . . . . . . . . . 17 |- 0 < _e
24	22, 23	gtneii 9764	. . . . . . . . . . . . . . . 16 |- _e =/= 0
25	24	a1i 11	. . . . . . . . . . . . . . 15 |- ( n e. NN -> _e =/= 0 )
26	15, 21, 25	divcld 10405	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( n / _e ) e. CC )
27	26, 10	expcld 12454	. . . . . . . . . . . . 13 |- ( n e. NN -> ( ( n / _e ) ^ n ) e. CC )
28	17, 27	mulcld 9681	. . . . . . . . . . . 12 |- ( n e. NN -> ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) e. CC )
29		2rp 11330	. . . . . . . . . . . . . . . . 17 |- 2 e. RR+
30	29	a1i 11	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> 2 e. RR+ )
31		nnrp 11334	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> n e. RR+ )
32	30, 31	rpmulcld 11380	. . . . . . . . . . . . . . 15 |- ( n e. NN -> ( 2 x. n ) e. RR+ )
33	32	sqrtgt0d 13551	. . . . . . . . . . . . . 14 |- ( n e. NN -> 0 < ( sqr ` ( 2 x. n ) ) )
34	33	gt0ne0d 10199	. . . . . . . . . . . . 13 |- ( n e. NN -> ( sqr ` ( 2 x. n ) ) =/= 0 )
35		nnne0 10664	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n =/= 0 )
36	15, 21, 35, 25	divne0d 10421	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( n / _e ) =/= 0 )
37		nnz 10983	. . . . . . . . . . . . . 14 |- ( n e. NN -> n e. ZZ )
38	26, 36, 37	expne0d 12460	. . . . . . . . . . . . 13 |- ( n e. NN -> ( ( n / _e ) ^ n ) =/= 0 )
39	17, 27, 34, 38	mulne0d 10286	. . . . . . . . . . . 12 |- ( n e. NN -> ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) =/= 0 )
40	13, 28, 39	divcld 10405	. . . . . . . . . . 11 |- ( n e. NN -> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) e. CC )
41	1	fvmpt2 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 ) ) ) )
42	40, 41	mpdan 681	. . . . . . . . . 10 |- ( n e. NN -> ( A ` n ) = ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) )
43	42, 40	eqeltrd 2549	. . . . . . . . 9 |- ( n e. NN -> ( A ` n ) e. CC )
44		nnne0 10664	. . . . . . . . . . . 12 |- ( ( ! ` n ) e. NN -> ( ! ` n ) =/= 0 )
45	10, 11, 44	3syl 18	. . . . . . . . . . 11 |- ( n e. NN -> ( ! ` n ) =/= 0 )
46	13, 28, 45, 39	divne0d 10421	. . . . . . . . . 10 |- ( n e. NN -> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) =/= 0 )
47	42, 46	eqnetrd 2710	. . . . . . . . 9 |- ( n e. NN -> ( A ` n ) =/= 0 )
48	43, 47	logcld 23599	. . . . . . . 8 |- ( n e. NN -> ( log ` ( A ` n ) ) e. CC )
49	2, 48	fmpti 6060	. . . . . . 7 |- B : NN --> CC
50	49	a1i 11	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> B : NN --> CC )
51		simpr 468	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> B ~~> d )
52	4	recnd 9687	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> d e. CC )
53	6, 7, 9, 50, 51, 52	climcncf 22010	. . . . 5 |- ( ( d e. RR /\ B ~~> d ) -> ( exp o. B ) ~~> ( exp ` d ) )
54	8	elexi 3041	. . . . . . . . 9 |- exp e. _V
55		nnex 10637	. . . . . . . . . . 11 |- NN e. _V
56	55	mptex 6152	. . . . . . . . . 10 |- ( n e. NN |-> ( log ` ( A ` n ) ) ) e. _V
57	2, 56	eqeltri 2545	. . . . . . . . 9 |- B e. _V
58	54, 57	coex 6764	. . . . . . . 8 |- ( exp o. B ) e. _V
59	58	a1i 11	. . . . . . 7 |- ( T. -> ( exp o. B ) e. _V )
60	55	mptex 6152	. . . . . . . . 9 |- ( n e. NN |-> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) ) e. _V
61	1, 60	eqeltri 2545	. . . . . . . 8 |- A e. _V
62	61	a1i 11	. . . . . . 7 |- ( T. -> A e. _V )
63		1zzd 10992	. . . . . . 7 |- ( T. -> 1 e. ZZ )
64	2	funmpt2 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 )
67	1	stirlinglem2 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 )
70	67, 68, 69	3syl 18	. . . . . . . . . . . . 13 |- ( n e. NN -> ( log ` ( A ` n ) ) e. _V )
71	66, 70	mprgbir 2771	. . . . . . . . . . . 12 |- NN = { n e. NN | ( log ` ( A ` n ) ) e. _V }
72	2	dmmpt 5337	. . . . . . . . . . . 12 |- dom B = { n e. NN | ( log ` ( A ` n ) ) e. _V }
73	71, 72	eqtr4i 2496	. . . . . . . . . . 11 |- NN = dom B
74	65, 73	syl6eleq 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 ) ) )
76	64, 74, 75	sylancr 676	. . . . . . . . 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 468	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN /\ n = k ) -> n = k )
79	78	fveq2d 5883	. . . . . . . . . . . . . . 15 |- ( ( k e. NN /\ n = k ) -> ( ! ` n ) = ( ! ` k ) )
80	78	oveq2d 6324	. . . . . . . . . . . . . . . . 17 |- ( ( k e. NN /\ n = k ) -> ( 2 x. n ) = ( 2 x. k ) )
81	80	fveq2d 5883	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN /\ n = k ) -> ( sqr ` ( 2 x. n ) ) = ( sqr ` ( 2 x. k ) ) )
82	78	oveq1d 6323	. . . . . . . . . . . . . . . . 17 |- ( ( k e. NN /\ n = k ) -> ( n / _e ) = ( k / _e ) )
83	82, 78	oveq12d 6326	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN /\ n = k ) -> ( ( n / _e ) ^ n ) = ( ( k / _e ) ^ k ) )
84	81, 83	oveq12d 6326	. . . . . . . . . . . . . . 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 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 )
89	86, 87, 88	3syl 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 )
92	90, 91	mulcld 9681	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> ( 2 x. k ) e. CC )
93	92	sqrtcld 13576	. . . . . . . . . . . . . . . 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 10405	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> ( k / _e ) e. CC )
97	96, 86	expcld 12454	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( ( k / _e ) ^ k ) e. CC )
98	93, 97	mulcld 9681	. . . . . . . . . . . . . . 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 11334	. . . . . . . . . . . . . . . . . . 19 |- ( k e. NN -> k e. RR+ )
101	99, 100	rpmulcld 11380	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> ( 2 x. k ) e. RR+ )
102	101	sqrtgt0d 13551	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> 0 < ( sqr ` ( 2 x. k ) ) )
103	102	gt0ne0d 10199	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( sqr ` ( 2 x. k ) ) =/= 0 )
104		nnne0 10664	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> k =/= 0 )
105	91, 94, 104, 95	divne0d 10421	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> ( k / _e ) =/= 0 )
106		nnz 10983	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> k e. ZZ )
107	96, 105, 106	expne0d 12460	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( ( k / _e ) ^ k ) =/= 0 )
108	93, 97, 103, 107	mulne0d 10286	. . . . . . . . . . . . . . 15 |- ( k e. NN -> ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) =/= 0 )
109	89, 98, 108	divcld 10405	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) e. CC )
110	77, 85, 65, 109	fvmptd 5969	. . . . . . . . . . . . 13 |- ( k e. NN -> ( A ` k ) = ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) )
111	110, 109	eqeltrd 2549	. . . . . . . . . . . 12 |- ( k e. NN -> ( A ` k ) e. CC )
112		nnne0 10664	. . . . . . . . . . . . . . 15 |- ( ( ! ` k ) e. NN -> ( ! ` k ) =/= 0 )
113	86, 87, 112	3syl 18	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( ! ` k ) =/= 0 )
114	89, 98, 113, 108	divne0d 10421	. . . . . . . . . . . . 13 |- ( k e. NN -> ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) =/= 0 )
115	110, 114	eqnetrd 2710	. . . . . . . . . . . 12 |- ( k e. NN -> ( A ` k ) =/= 0 )
116	111, 115	logcld 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 ) ) ) )
120	1, 119	nfcxfr 2610	. . . . . . . . . . . . . 14 |- F/_ n A
121	120, 117	nffv 5886	. . . . . . . . . . . . 13 |- F/_ n ( A ` k )
122	118, 121	nffv 5886	. . . . . . . . . . . 12 |- F/_ n ( log ` ( A ` k ) )
123		fveq2 5879	. . . . . . . . . . . . 13 |- ( n = k -> ( A ` n ) = ( A ` k ) )
124	123	fveq2d 5883	. . . . . . . . . . . 12 |- ( n = k -> ( log ` ( A ` n ) ) = ( log ` ( A ` k ) ) )
125	117, 122, 124, 2	fvmptf 5981	. . . . . . . . . . 11 |- ( ( k e. NN /\ ( log ` ( A ` k ) ) e. CC ) -> ( B ` k ) = ( log ` ( A ` k ) ) )
126	116, 125	mpdan 681	. . . . . . . . . 10 |- ( k e. NN -> ( B ` k ) = ( log ` ( A ` k ) ) )
127	126	fveq2d 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 ) )
129	111, 115, 128	syl2anc 673	. . . . . . . . 9 |- ( k e. NN -> ( exp ` ( log ` ( A ` k ) ) ) = ( A ` k ) )
130	76, 127, 129	3eqtrd 2509	. . . . . . . 8 |- ( k e. NN -> ( ( exp o. B ) ` k ) = ( A ` k ) )
131	130	adantl 473	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> ( ( exp o. B ) ` k ) = ( A ` k ) )
132	6, 59, 62, 63, 131	climeq 13708	. . . . . 6 |- ( T. -> ( ( exp o. B ) ~~> ( exp ` d ) <-> A ~~> ( exp ` d ) ) )
133	132	trud 1461	. . . . 5 |- ( ( exp o. B ) ~~> ( exp ` d ) <-> A ~~> ( exp ` d ) )
134	53, 133	sylib 201	. . . 4 |- ( ( d e. RR /\ B ~~> d ) -> A ~~> ( exp ` d ) )
135		breq2 4399	. . . . 5 |- ( c = ( exp ` d ) -> ( A ~~> c <-> A ~~> ( exp ` d ) ) )
136	135	rspcev 3136	. . . 4 |- ( ( ( exp ` d ) e. RR+ /\ A ~~> ( exp ` d ) ) -> E. c e. RR+ A ~~> c )
137	5, 134, 136	syl2anc 673	. . 3 |- ( ( d e. RR /\ B ~~> d ) -> E. c e. RR+ A ~~> c )
138	137	rexlimiva 2868	. 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 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