Theorem stirlinglem14 40304

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 k d 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 40303	. 2 |- E. d e. RR B ~~> d
4		simpl 473	. . . . 5 |- ( ( d e. RR /\ B ~~> d ) -> d e. RR )
5	4	rpefcld 14835	. . . 4 |- ( ( d e. RR /\ B ~~> d ) -> ( exp ` d ) e. RR+ )
6		nnuz 11723	. . . . . 6 |- NN = ( ZZ>= ` 1 )
7		1zzd 11408	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> 1 e. ZZ )
8		efcn 24197	. . . . . . 7 |- exp e. ( CC -cn-> CC )
9	8	a1i 11	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> exp e. ( CC -cn-> CC ) )
10		nnnn0 11299	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. NN0 )
11		faccl 13070	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( ! ` n ) e. NN )
12		nncn 11028	. . . . . . . . . . . . 13 |- ( ( ! ` n ) e. NN -> ( ! ` n ) e. CC )
13	10, 11, 12	3syl 18	. . . . . . . . . . . 12 |- ( n e. NN -> ( ! ` n ) e. CC )
14		2cnd 11093	. . . . . . . . . . . . . . 15 |- ( n e. NN -> 2 e. CC )
15		nncn 11028	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n e. CC )
16	14, 15	mulcld 10060	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( 2 x. n ) e. CC )
17	16	sqrtcld 14176	. . . . . . . . . . . . 13 |- ( n e. NN -> ( sqr ` ( 2 x. n ) ) e. CC )
18		epr 14936	. . . . . . . . . . . . . . . . 17 |- _e e. RR+
19		rpcn 11841	. . . . . . . . . . . . . . . . 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 10040	. . . . . . . . . . . . . . . . 17 |- 0 e. RR
23		epos 14935	. . . . . . . . . . . . . . . . 17 |- 0 < _e
24	22, 23	gtneii 10149	. . . . . . . . . . . . . . . 16 |- _e =/= 0
25	24	a1i 11	. . . . . . . . . . . . . . 15 |- ( n e. NN -> _e =/= 0 )
26	15, 21, 25	divcld 10801	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( n / _e ) e. CC )
27	26, 10	expcld 13008	. . . . . . . . . . . . 13 |- ( n e. NN -> ( ( n / _e ) ^ n ) e. CC )
28	17, 27	mulcld 10060	. . . . . . . . . . . 12 |- ( n e. NN -> ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) e. CC )
29		2rp 11837	. . . . . . . . . . . . . . . . 17 |- 2 e. RR+
30	29	a1i 11	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> 2 e. RR+ )
31		nnrp 11842	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> n e. RR+ )
32	30, 31	rpmulcld 11888	. . . . . . . . . . . . . . 15 |- ( n e. NN -> ( 2 x. n ) e. RR+ )
33	32	sqrtgt0d 14151	. . . . . . . . . . . . . 14 |- ( n e. NN -> 0 < ( sqr ` ( 2 x. n ) ) )
34	33	gt0ne0d 10592	. . . . . . . . . . . . 13 |- ( n e. NN -> ( sqr ` ( 2 x. n ) ) =/= 0 )
35		nnne0 11053	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n =/= 0 )
36	15, 21, 35, 25	divne0d 10817	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( n / _e ) =/= 0 )
37		nnz 11399	. . . . . . . . . . . . . 14 |- ( n e. NN -> n e. ZZ )
38	26, 36, 37	expne0d 13014	. . . . . . . . . . . . 13 |- ( n e. NN -> ( ( n / _e ) ^ n ) =/= 0 )
39	17, 27, 34, 38	mulne0d 10679	. . . . . . . . . . . 12 |- ( n e. NN -> ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) =/= 0 )
40	13, 28, 39	divcld 10801	. . . . . . . . . . 11 |- ( n e. NN -> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) e. CC )
41	1	fvmpt2 6291	. . . . . . . . . . 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 702	. . . . . . . . . 10 |- ( n e. NN -> ( A ` n ) = ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) )
43	42, 40	eqeltrd 2701	. . . . . . . . 9 |- ( n e. NN -> ( A ` n ) e. CC )
44		nnne0 11053	. . . . . . . . . . . 12 |- ( ( ! ` n ) e. NN -> ( ! ` n ) =/= 0 )
45	10, 11, 44	3syl 18	. . . . . . . . . . 11 |- ( n e. NN -> ( ! ` n ) =/= 0 )
46	13, 28, 45, 39	divne0d 10817	. . . . . . . . . 10 |- ( n e. NN -> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) =/= 0 )
47	42, 46	eqnetrd 2861	. . . . . . . . 9 |- ( n e. NN -> ( A ` n ) =/= 0 )
48	43, 47	logcld 24317	. . . . . . . 8 |- ( n e. NN -> ( log ` ( A ` n ) ) e. CC )
49	2, 48	fmpti 6383	. . . . . . 7 |- B : NN --> CC
50	49	a1i 11	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> B : NN --> CC )
51		simpr 477	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> B ~~> d )
52	4	recnd 10068	. . . . . 6 |- ( ( d e. RR /\ B ~~> d ) -> d e. CC )
53	6, 7, 9, 50, 51, 52	climcncf 22703	. . . . 5 |- ( ( d e. RR /\ B ~~> d ) -> ( exp o. B ) ~~> ( exp ` d ) )
54	8	elexi 3213	. . . . . . . . 9 |- exp e. _V
55		nnex 11026	. . . . . . . . . . 11 |- NN e. _V
56	55	mptex 6486	. . . . . . . . . 10 |- ( n e. NN |-> ( log ` ( A ` n ) ) ) e. _V
57	2, 56	eqeltri 2697	. . . . . . . . 9 |- B e. _V
58	54, 57	coex 7118	. . . . . . . 8 |- ( exp o. B ) e. _V
59	58	a1i 11	. . . . . . 7 |- ( T. -> ( exp o. B ) e. _V )
60	55	mptex 6486	. . . . . . . . 9 |- ( n e. NN |-> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) ) e. _V
61	1, 60	eqeltri 2697	. . . . . . . 8 |- A e. _V
62	61	a1i 11	. . . . . . 7 |- ( T. -> A e. _V )
63		1zzd 11408	. . . . . . 7 |- ( T. -> 1 e. ZZ )
64	2	funmpt2 5927	. . . . . . . . . 10 |- Fun B
65		id 22	. . . . . . . . . . 11 |- ( k e. NN -> k e. NN )
66		rabid2 3118	. . . . . . . . . . . . 13 |- ( NN = { n e. NN | ( log ` ( A ` n ) ) e. _V } <-> A. n e. NN ( log ` ( A ` n ) ) e. _V )
67	1	stirlinglem2 40292	. . . . . . . . . . . . . 14 |- ( n e. NN -> ( A ` n ) e. RR+ )
68		relogcl 24322	. . . . . . . . . . . . . 14 |- ( ( A ` n ) e. RR+ -> ( log ` ( A ` n ) ) e. RR )
69		elex 3212	. . . . . . . . . . . . . 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 2927	. . . . . . . . . . . 12 |- NN = { n e. NN | ( log ` ( A ` n ) ) e. _V }
72	2	dmmpt 5630	. . . . . . . . . . . 12 |- dom B = { n e. NN | ( log ` ( A ` n ) ) e. _V }
73	71, 72	eqtr4i 2647	. . . . . . . . . . 11 |- NN = dom B
74	65, 73	syl6eleq 2711	. . . . . . . . . 10 |- ( k e. NN -> k e. dom B )
75		fvco 6274	. . . . . . . . . 10 |- ( ( Fun B /\ k e. dom B ) -> ( ( exp o. B ) ` k ) = ( exp ` ( B ` k ) ) )
76	64, 74, 75	sylancr 695	. . . . . . . . 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 477	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN /\ n = k ) -> n = k )
79	78	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( k e. NN /\ n = k ) -> ( ! ` n ) = ( ! ` k ) )
80	78	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( ( k e. NN /\ n = k ) -> ( 2 x. n ) = ( 2 x. k ) )
81	80	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN /\ n = k ) -> ( sqr ` ( 2 x. n ) ) = ( sqr ` ( 2 x. k ) ) )
82	78	oveq1d 6665	. . . . . . . . . . . . . . . . 17 |- ( ( k e. NN /\ n = k ) -> ( n / _e ) = ( k / _e ) )
83	82, 78	oveq12d 6668	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN /\ n = k ) -> ( ( n / _e ) ^ n ) = ( ( k / _e ) ^ k ) )
84	81, 83	oveq12d 6668	. . . . . . . . . . . . . . 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 6668	. . . . . . . . . . . . . 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 11299	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> k e. NN0 )
87		faccl 13070	. . . . . . . . . . . . . . . 16 |- ( k e. NN0 -> ( ! ` k ) e. NN )
88		nncn 11028	. . . . . . . . . . . . . . . 16 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
89	86, 87, 88	3syl 18	. . . . . . . . . . . . . . 15 |- ( k e. NN -> ( ! ` k ) e. CC )
90		2cnd 11093	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> 2 e. CC )
91		nncn 11028	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> k e. CC )
92	90, 91	mulcld 10060	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> ( 2 x. k ) e. CC )
93	92	sqrtcld 14176	. . . . . . . . . . . . . . . 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 10801	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> ( k / _e ) e. CC )
97	96, 86	expcld 13008	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( ( k / _e ) ^ k ) e. CC )
98	93, 97	mulcld 10060	. . . . . . . . . . . . . . 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 11842	. . . . . . . . . . . . . . . . . . 19 |- ( k e. NN -> k e. RR+ )
101	99, 100	rpmulcld 11888	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> ( 2 x. k ) e. RR+ )
102	101	sqrtgt0d 14151	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> 0 < ( sqr ` ( 2 x. k ) ) )
103	102	gt0ne0d 10592	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( sqr ` ( 2 x. k ) ) =/= 0 )
104		nnne0 11053	. . . . . . . . . . . . . . . . . 18 |- ( k e. NN -> k =/= 0 )
105	91, 94, 104, 95	divne0d 10817	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> ( k / _e ) =/= 0 )
106		nnz 11399	. . . . . . . . . . . . . . . . 17 |- ( k e. NN -> k e. ZZ )
107	96, 105, 106	expne0d 13014	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( ( k / _e ) ^ k ) =/= 0 )
108	93, 97, 103, 107	mulne0d 10679	. . . . . . . . . . . . . . 15 |- ( k e. NN -> ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) =/= 0 )
109	89, 98, 108	divcld 10801	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) e. CC )
110	77, 85, 65, 109	fvmptd 6288	. . . . . . . . . . . . 13 |- ( k e. NN -> ( A ` k ) = ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) )
111	110, 109	eqeltrd 2701	. . . . . . . . . . . 12 |- ( k e. NN -> ( A ` k ) e. CC )
112		nnne0 11053	. . . . . . . . . . . . . . 15 |- ( ( ! ` k ) e. NN -> ( ! ` k ) =/= 0 )
113	86, 87, 112	3syl 18	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( ! ` k ) =/= 0 )
114	89, 98, 113, 108	divne0d 10817	. . . . . . . . . . . . 13 |- ( k e. NN -> ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) =/= 0 )
115	110, 114	eqnetrd 2861	. . . . . . . . . . . 12 |- ( k e. NN -> ( A ` k ) =/= 0 )
116	111, 115	logcld 24317	. . . . . . . . . . 11 |- ( k e. NN -> ( log ` ( A ` k ) ) e. CC )
117		nfcv 2764	. . . . . . . . . . . 12 |- F/_ n k
118		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ n log
119		nfmpt1 4747	. . . . . . . . . . . . . . 15 |- F/_ n ( n e. NN |-> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) )
120	1, 119	nfcxfr 2762	. . . . . . . . . . . . . 14 |- F/_ n A
121	120, 117	nffv 6198	. . . . . . . . . . . . 13 |- F/_ n ( A ` k )
122	118, 121	nffv 6198	. . . . . . . . . . . 12 |- F/_ n ( log ` ( A ` k ) )
123		fveq2 6191	. . . . . . . . . . . . 13 |- ( n = k -> ( A ` n ) = ( A ` k ) )
124	123	fveq2d 6195	. . . . . . . . . . . 12 |- ( n = k -> ( log ` ( A ` n ) ) = ( log ` ( A ` k ) ) )
125	117, 122, 124, 2	fvmptf 6301	. . . . . . . . . . 11 |- ( ( k e. NN /\ ( log ` ( A ` k ) ) e. CC ) -> ( B ` k ) = ( log ` ( A ` k ) ) )
126	116, 125	mpdan 702	. . . . . . . . . 10 |- ( k e. NN -> ( B ` k ) = ( log ` ( A ` k ) ) )
127	126	fveq2d 6195	. . . . . . . . 9 |- ( k e. NN -> ( exp ` ( B ` k ) ) = ( exp ` ( log ` ( A ` k ) ) ) )
128		eflog 24323	. . . . . . . . . 10 |- ( ( ( A ` k ) e. CC /\ ( A ` k ) =/= 0 ) -> ( exp ` ( log ` ( A ` k ) ) ) = ( A ` k ) )
129	111, 115, 128	syl2anc 693	. . . . . . . . 9 |- ( k e. NN -> ( exp ` ( log ` ( A ` k ) ) ) = ( A ` k ) )
130	76, 127, 129	3eqtrd 2660	. . . . . . . 8 |- ( k e. NN -> ( ( exp o. B ) ` k ) = ( A ` k ) )
131	130	adantl 482	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> ( ( exp o. B ) ` k ) = ( A ` k ) )
132	6, 59, 62, 63, 131	climeq 14298	. . . . . 6 |- ( T. -> ( ( exp o. B ) ~~> ( exp ` d ) <-> A ~~> ( exp ` d ) ) )
133	132	trud 1493	. . . . 5 |- ( ( exp o. B ) ~~> ( exp ` d ) <-> A ~~> ( exp ` d ) )
134	53, 133	sylib 208	. . . 4 |- ( ( d e. RR /\ B ~~> d ) -> A ~~> ( exp ` d ) )
135		breq2 4657	. . . . 5 |- ( c = ( exp ` d ) -> ( A ~~> c <-> A ~~> ( exp ` d ) ) )
136	135	rspcev 3309	. . . 4 |- ( ( ( exp ` d ) e. RR+ /\ A ~~> ( exp ` d ) ) -> E. c e. RR+ A ~~> c )
137	5, 134, 136	syl2anc 693	. . 3 |- ( ( d e. RR /\ B ~~> d ) -> E. c e. RR+ A ~~> c )
138	137	rexlimiva 3028	. 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 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   RR+crp 11832   ^cexp 12860   !cfa 13060   sqrcsqrt 13973   ~~> cli 14215   expce 14792   _eceu 14793   -cn->ccncf 22679   logclog 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-dvds 14984  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-ulm 24131  df-log 24303  df-cxp 24304

This theorem is referenced by:  stirling  40306