Theorem logtayl2 24408

Description: Power series expression for the logarithm. (Contributed by Mario Carneiro, 31-Mar-2015.)

Hypothesis

Ref	Expression
logtayl2.s	|- S = ( 1 ( ball ` ( abs o. - ) ) 1 )

Assertion

Ref	Expression
logtayl2	|- ( A e. S -> seq 1 ( + , ( k e. NN |-> ( ( ( -u 1 ^ ( k - 1 ) ) / k ) x. ( ( A - 1 ) ^ k ) ) ) ) ~~> ( log ` A ) )

Distinct variable group:   A, k

Allowed substitution hint:   S( k)

Proof of Theorem logtayl2

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11723	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 11408	. . 3 |- ( A e. S -> 1 e. ZZ )
3		neg1cn 11124	. . . 4 |- -u 1 e. CC
4	3	a1i 11	. . 3 |- ( A e. S -> -u 1 e. CC )
5		ax-1cn 9994	. . . . . 6 |- 1 e. CC
6		logtayl2.s	. . . . . . . . 9 |- S = ( 1 ( ball ` ( abs o. - ) ) 1 )
7	6	eleq2i 2693	. . . . . . . 8 |- ( A e. S <-> A e. ( 1 ( ball ` ( abs o. - ) ) 1 ) )
8		cnxmet 22576	. . . . . . . . 9 |- ( abs o. - ) e. ( *Met ` CC )
9		1rp 11836	. . . . . . . . . 10 |- 1 e. RR+
10		rpxr 11840	. . . . . . . . . 10 |- ( 1 e. RR+ -> 1 e. RR* )
11	9, 10	ax-mp 5	. . . . . . . . 9 |- 1 e. RR*
12		elbl 22193	. . . . . . . . 9 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 1 e. CC /\ 1 e. RR* ) -> ( A e. ( 1 ( ball ` ( abs o. - ) ) 1 ) <-> ( A e. CC /\ ( 1 ( abs o. - ) A ) < 1 ) ) )
13	8, 5, 11, 12	mp3an 1424	. . . . . . . 8 |- ( A e. ( 1 ( ball ` ( abs o. - ) ) 1 ) <-> ( A e. CC /\ ( 1 ( abs o. - ) A ) < 1 ) )
14	7, 13	bitri 264	. . . . . . 7 |- ( A e. S <-> ( A e. CC /\ ( 1 ( abs o. - ) A ) < 1 ) )
15	14	simplbi 476	. . . . . 6 |- ( A e. S -> A e. CC )
16		subcl 10280	. . . . . 6 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
17	5, 15, 16	sylancr 695	. . . . 5 |- ( A e. S -> ( 1 - A ) e. CC )
18		eqid 2622	. . . . . . . 8 |- ( abs o. - ) = ( abs o. - )
19	18	cnmetdval 22574	. . . . . . 7 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 ( abs o. - ) A ) = ( abs ` ( 1 - A ) ) )
20	5, 15, 19	sylancr 695	. . . . . 6 |- ( A e. S -> ( 1 ( abs o. - ) A ) = ( abs ` ( 1 - A ) ) )
21	14	simprbi 480	. . . . . 6 |- ( A e. S -> ( 1 ( abs o. - ) A ) < 1 )
22	20, 21	eqbrtrrd 4677	. . . . 5 |- ( A e. S -> ( abs ` ( 1 - A ) ) < 1 )
23		logtayl 24406	. . . . 5 |- ( ( ( 1 - A ) e. CC /\ ( abs ` ( 1 - A ) ) < 1 ) -> seq 1 ( + , ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) ) ) ~~> -u ( log ` ( 1 - ( 1 - A ) ) ) )
24	17, 22, 23	syl2anc 693	. . . 4 |- ( A e. S -> seq 1 ( + , ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) ) ) ~~> -u ( log ` ( 1 - ( 1 - A ) ) ) )
25		nncan 10310	. . . . . . 7 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - ( 1 - A ) ) = A )
26	5, 15, 25	sylancr 695	. . . . . 6 |- ( A e. S -> ( 1 - ( 1 - A ) ) = A )
27	26	fveq2d 6195	. . . . 5 |- ( A e. S -> ( log ` ( 1 - ( 1 - A ) ) ) = ( log ` A ) )
28	27	negeqd 10275	. . . 4 |- ( A e. S -> -u ( log ` ( 1 - ( 1 - A ) ) ) = -u ( log ` A ) )
29	24, 28	breqtrd 4679	. . 3 |- ( A e. S -> seq 1 ( + , ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) ) ) ~~> -u ( log ` A ) )
30		oveq2 6658	. . . . . . 7 |- ( k = n -> ( ( 1 - A ) ^ k ) = ( ( 1 - A ) ^ n ) )
31		id 22	. . . . . . 7 |- ( k = n -> k = n )
32	30, 31	oveq12d 6668	. . . . . 6 |- ( k = n -> ( ( ( 1 - A ) ^ k ) / k ) = ( ( ( 1 - A ) ^ n ) / n ) )
33		eqid 2622	. . . . . 6 |- ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) ) = ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) )
34		ovex 6678	. . . . . 6 |- ( ( ( 1 - A ) ^ n ) / n ) e. _V
35	32, 33, 34	fvmpt 6282	. . . . 5 |- ( n e. NN -> ( ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) ) ` n ) = ( ( ( 1 - A ) ^ n ) / n ) )
36	35	adantl 482	. . . 4 |- ( ( A e. S /\ n e. NN ) -> ( ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) ) ` n ) = ( ( ( 1 - A ) ^ n ) / n ) )
37		nnnn0 11299	. . . . . 6 |- ( n e. NN -> n e. NN0 )
38		expcl 12878	. . . . . 6 |- ( ( ( 1 - A ) e. CC /\ n e. NN0 ) -> ( ( 1 - A ) ^ n ) e. CC )
39	17, 37, 38	syl2an 494	. . . . 5 |- ( ( A e. S /\ n e. NN ) -> ( ( 1 - A ) ^ n ) e. CC )
40		nncn 11028	. . . . . 6 |- ( n e. NN -> n e. CC )
41	40	adantl 482	. . . . 5 |- ( ( A e. S /\ n e. NN ) -> n e. CC )
42		nnne0 11053	. . . . . 6 |- ( n e. NN -> n =/= 0 )
43	42	adantl 482	. . . . 5 |- ( ( A e. S /\ n e. NN ) -> n =/= 0 )
44	39, 41, 43	divcld 10801	. . . 4 |- ( ( A e. S /\ n e. NN ) -> ( ( ( 1 - A ) ^ n ) / n ) e. CC )
45	36, 44	eqeltrd 2701	. . 3 |- ( ( A e. S /\ n e. NN ) -> ( ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) ) ` n ) e. CC )
46	39, 41, 43	divnegd 10814	. . . . . 6 |- ( ( A e. S /\ n e. NN ) -> -u ( ( ( 1 - A ) ^ n ) / n ) = ( -u ( ( 1 - A ) ^ n ) / n ) )
47	44	mulm1d 10482	. . . . . 6 |- ( ( A e. S /\ n e. NN ) -> ( -u 1 x. ( ( ( 1 - A ) ^ n ) / n ) ) = -u ( ( ( 1 - A ) ^ n ) / n ) )
48	37	adantl 482	. . . . . . . . . 10 |- ( ( A e. S /\ n e. NN ) -> n e. NN0 )
49		expcl 12878	. . . . . . . . . 10 |- ( ( -u 1 e. CC /\ n e. NN0 ) -> ( -u 1 ^ n ) e. CC )
50	3, 48, 49	sylancr 695	. . . . . . . . 9 |- ( ( A e. S /\ n e. NN ) -> ( -u 1 ^ n ) e. CC )
51		subcl 10280	. . . . . . . . . . 11 |- ( ( A e. CC /\ 1 e. CC ) -> ( A - 1 ) e. CC )
52	15, 5, 51	sylancl 694	. . . . . . . . . 10 |- ( A e. S -> ( A - 1 ) e. CC )
53		expcl 12878	. . . . . . . . . 10 |- ( ( ( A - 1 ) e. CC /\ n e. NN0 ) -> ( ( A - 1 ) ^ n ) e. CC )
54	52, 37, 53	syl2an 494	. . . . . . . . 9 |- ( ( A e. S /\ n e. NN ) -> ( ( A - 1 ) ^ n ) e. CC )
55	50, 54	mulneg1d 10483	. . . . . . . 8 |- ( ( A e. S /\ n e. NN ) -> ( -u ( -u 1 ^ n ) x. ( ( A - 1 ) ^ n ) ) = -u ( ( -u 1 ^ n ) x. ( ( A - 1 ) ^ n ) ) )
56	3	a1i 11	. . . . . . . . . . 11 |- ( ( A e. S /\ n e. NN ) -> -u 1 e. CC )
57		neg1ne0 11126	. . . . . . . . . . . 12 |- -u 1 =/= 0
58	57	a1i 11	. . . . . . . . . . 11 |- ( ( A e. S /\ n e. NN ) -> -u 1 =/= 0 )
59		nnz 11399	. . . . . . . . . . . 12 |- ( n e. NN -> n e. ZZ )
60	59	adantl 482	. . . . . . . . . . 11 |- ( ( A e. S /\ n e. NN ) -> n e. ZZ )
61	56, 58, 60	expm1d 13018	. . . . . . . . . 10 |- ( ( A e. S /\ n e. NN ) -> ( -u 1 ^ ( n - 1 ) ) = ( ( -u 1 ^ n ) / -u 1 ) )
62	5	a1i 11	. . . . . . . . . . 11 |- ( ( A e. S /\ n e. NN ) -> 1 e. CC )
63		ax-1ne0 10005	. . . . . . . . . . . 12 |- 1 =/= 0
64	63	a1i 11	. . . . . . . . . . 11 |- ( ( A e. S /\ n e. NN ) -> 1 =/= 0 )
65	50, 62, 64	divneg2d 10815	. . . . . . . . . 10 |- ( ( A e. S /\ n e. NN ) -> -u ( ( -u 1 ^ n ) / 1 ) = ( ( -u 1 ^ n ) / -u 1 ) )
66	50	div1d 10793	. . . . . . . . . . 11 |- ( ( A e. S /\ n e. NN ) -> ( ( -u 1 ^ n ) / 1 ) = ( -u 1 ^ n ) )
67	66	negeqd 10275	. . . . . . . . . 10 |- ( ( A e. S /\ n e. NN ) -> -u ( ( -u 1 ^ n ) / 1 ) = -u ( -u 1 ^ n ) )
68	61, 65, 67	3eqtr2d 2662	. . . . . . . . 9 |- ( ( A e. S /\ n e. NN ) -> ( -u 1 ^ ( n - 1 ) ) = -u ( -u 1 ^ n ) )
69	68	oveq1d 6665	. . . . . . . 8 |- ( ( A e. S /\ n e. NN ) -> ( ( -u 1 ^ ( n - 1 ) ) x. ( ( A - 1 ) ^ n ) ) = ( -u ( -u 1 ^ n ) x. ( ( A - 1 ) ^ n ) ) )
70	52	mulm1d 10482	. . . . . . . . . . . . 13 |- ( A e. S -> ( -u 1 x. ( A - 1 ) ) = -u ( A - 1 ) )
71		negsubdi2 10340	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ 1 e. CC ) -> -u ( A - 1 ) = ( 1 - A ) )
72	15, 5, 71	sylancl 694	. . . . . . . . . . . . 13 |- ( A e. S -> -u ( A - 1 ) = ( 1 - A ) )
73	70, 72	eqtr2d 2657	. . . . . . . . . . . 12 |- ( A e. S -> ( 1 - A ) = ( -u 1 x. ( A - 1 ) ) )
74	73	oveq1d 6665	. . . . . . . . . . 11 |- ( A e. S -> ( ( 1 - A ) ^ n ) = ( ( -u 1 x. ( A - 1 ) ) ^ n ) )
75	74	adantr 481	. . . . . . . . . 10 |- ( ( A e. S /\ n e. NN ) -> ( ( 1 - A ) ^ n ) = ( ( -u 1 x. ( A - 1 ) ) ^ n ) )
76		mulexp 12899	. . . . . . . . . . . 12 |- ( ( -u 1 e. CC /\ ( A - 1 ) e. CC /\ n e. NN0 ) -> ( ( -u 1 x. ( A - 1 ) ) ^ n ) = ( ( -u 1 ^ n ) x. ( ( A - 1 ) ^ n ) ) )
77	3, 76	mp3an1 1411	. . . . . . . . . . 11 |- ( ( ( A - 1 ) e. CC /\ n e. NN0 ) -> ( ( -u 1 x. ( A - 1 ) ) ^ n ) = ( ( -u 1 ^ n ) x. ( ( A - 1 ) ^ n ) ) )
78	52, 37, 77	syl2an 494	. . . . . . . . . 10 |- ( ( A e. S /\ n e. NN ) -> ( ( -u 1 x. ( A - 1 ) ) ^ n ) = ( ( -u 1 ^ n ) x. ( ( A - 1 ) ^ n ) ) )
79	75, 78	eqtrd 2656	. . . . . . . . 9 |- ( ( A e. S /\ n e. NN ) -> ( ( 1 - A ) ^ n ) = ( ( -u 1 ^ n ) x. ( ( A - 1 ) ^ n ) ) )
80	79	negeqd 10275	. . . . . . . 8 |- ( ( A e. S /\ n e. NN ) -> -u ( ( 1 - A ) ^ n ) = -u ( ( -u 1 ^ n ) x. ( ( A - 1 ) ^ n ) ) )
81	55, 69, 80	3eqtr4d 2666	. . . . . . 7 |- ( ( A e. S /\ n e. NN ) -> ( ( -u 1 ^ ( n - 1 ) ) x. ( ( A - 1 ) ^ n ) ) = -u ( ( 1 - A ) ^ n ) )
82	81	oveq1d 6665	. . . . . 6 |- ( ( A e. S /\ n e. NN ) -> ( ( ( -u 1 ^ ( n - 1 ) ) x. ( ( A - 1 ) ^ n ) ) / n ) = ( -u ( ( 1 - A ) ^ n ) / n ) )
83	46, 47, 82	3eqtr4d 2666	. . . . 5 |- ( ( A e. S /\ n e. NN ) -> ( -u 1 x. ( ( ( 1 - A ) ^ n ) / n ) ) = ( ( ( -u 1 ^ ( n - 1 ) ) x. ( ( A - 1 ) ^ n ) ) / n ) )
84		nnm1nn0 11334	. . . . . . . 8 |- ( n e. NN -> ( n - 1 ) e. NN0 )
85	84	adantl 482	. . . . . . 7 |- ( ( A e. S /\ n e. NN ) -> ( n - 1 ) e. NN0 )
86		expcl 12878	. . . . . . 7 |- ( ( -u 1 e. CC /\ ( n - 1 ) e. NN0 ) -> ( -u 1 ^ ( n - 1 ) ) e. CC )
87	3, 85, 86	sylancr 695	. . . . . 6 |- ( ( A e. S /\ n e. NN ) -> ( -u 1 ^ ( n - 1 ) ) e. CC )
88	87, 54, 41, 43	div23d 10838	. . . . 5 |- ( ( A e. S /\ n e. NN ) -> ( ( ( -u 1 ^ ( n - 1 ) ) x. ( ( A - 1 ) ^ n ) ) / n ) = ( ( ( -u 1 ^ ( n - 1 ) ) / n ) x. ( ( A - 1 ) ^ n ) ) )
89	83, 88	eqtr2d 2657	. . . 4 |- ( ( A e. S /\ n e. NN ) -> ( ( ( -u 1 ^ ( n - 1 ) ) / n ) x. ( ( A - 1 ) ^ n ) ) = ( -u 1 x. ( ( ( 1 - A ) ^ n ) / n ) ) )
90		oveq1 6657	. . . . . . . . 9 |- ( k = n -> ( k - 1 ) = ( n - 1 ) )
91	90	oveq2d 6666	. . . . . . . 8 |- ( k = n -> ( -u 1 ^ ( k - 1 ) ) = ( -u 1 ^ ( n - 1 ) ) )
92	91, 31	oveq12d 6668	. . . . . . 7 |- ( k = n -> ( ( -u 1 ^ ( k - 1 ) ) / k ) = ( ( -u 1 ^ ( n - 1 ) ) / n ) )
93		oveq2 6658	. . . . . . 7 |- ( k = n -> ( ( A - 1 ) ^ k ) = ( ( A - 1 ) ^ n ) )
94	92, 93	oveq12d 6668	. . . . . 6 |- ( k = n -> ( ( ( -u 1 ^ ( k - 1 ) ) / k ) x. ( ( A - 1 ) ^ k ) ) = ( ( ( -u 1 ^ ( n - 1 ) ) / n ) x. ( ( A - 1 ) ^ n ) ) )
95		eqid 2622	. . . . . 6 |- ( k e. NN |-> ( ( ( -u 1 ^ ( k - 1 ) ) / k ) x. ( ( A - 1 ) ^ k ) ) ) = ( k e. NN |-> ( ( ( -u 1 ^ ( k - 1 ) ) / k ) x. ( ( A - 1 ) ^ k ) ) )
96		ovex 6678	. . . . . 6 |- ( ( ( -u 1 ^ ( n - 1 ) ) / n ) x. ( ( A - 1 ) ^ n ) ) e. _V
97	94, 95, 96	fvmpt 6282	. . . . 5 |- ( n e. NN -> ( ( k e. NN |-> ( ( ( -u 1 ^ ( k - 1 ) ) / k ) x. ( ( A - 1 ) ^ k ) ) ) ` n ) = ( ( ( -u 1 ^ ( n - 1 ) ) / n ) x. ( ( A - 1 ) ^ n ) ) )
98	97	adantl 482	. . . 4 |- ( ( A e. S /\ n e. NN ) -> ( ( k e. NN |-> ( ( ( -u 1 ^ ( k - 1 ) ) / k ) x. ( ( A - 1 ) ^ k ) ) ) ` n ) = ( ( ( -u 1 ^ ( n - 1 ) ) / n ) x. ( ( A - 1 ) ^ n ) ) )
99	36	oveq2d 6666	. . . 4 |- ( ( A e. S /\ n e. NN ) -> ( -u 1 x. ( ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) ) ` n ) ) = ( -u 1 x. ( ( ( 1 - A ) ^ n ) / n ) ) )
100	89, 98, 99	3eqtr4d 2666	. . 3 |- ( ( A e. S /\ n e. NN ) -> ( ( k e. NN |-> ( ( ( -u 1 ^ ( k - 1 ) ) / k ) x. ( ( A - 1 ) ^ k ) ) ) ` n ) = ( -u 1 x. ( ( k e. NN |-> ( ( ( 1 - A ) ^ k ) / k ) ) ` n ) ) )
101	1, 2, 4, 29, 45, 100	isermulc2 14388	. 2 |- ( A e. S -> seq 1 ( + , ( k e. NN |-> ( ( ( -u 1 ^ ( k - 1 ) ) / k ) x. ( ( A - 1 ) ^ k ) ) ) ) ~~> ( -u 1 x. -u ( log ` A ) ) )
102	6	dvlog2lem 24398	. . . . . . . 8 |- S C_ ( CC \ ( -oo (,] 0 ) )
103	102	sseli 3599	. . . . . . 7 |- ( A e. S -> A e. ( CC \ ( -oo (,] 0 ) ) )
104		eqid 2622	. . . . . . . 8 |- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) )
105	104	logdmn0 24386	. . . . . . 7 |- ( A e. ( CC \ ( -oo (,] 0 ) ) -> A =/= 0 )
106	103, 105	syl 17	. . . . . 6 |- ( A e. S -> A =/= 0 )
107	15, 106	logcld 24317	. . . . 5 |- ( A e. S -> ( log ` A ) e. CC )
108	107	negcld 10379	. . . 4 |- ( A e. S -> -u ( log ` A ) e. CC )
109	108	mulm1d 10482	. . 3 |- ( A e. S -> ( -u 1 x. -u ( log ` A ) ) = -u -u ( log ` A ) )
110	107	negnegd 10383	. . 3 |- ( A e. S -> -u -u ( log ` A ) = ( log ` A ) )
111	109, 110	eqtrd 2656	. 2 |- ( A e. S -> ( -u 1 x. -u ( log ` A ) ) = ( log ` A ) )
112	101, 111	breqtrd 4679	1 |- ( A e. S -> seq 1 ( + , ( k e. NN |-> ( ( ( -u 1 ^ ( k - 1 ) ) / k ) x. ( ( A - 1 ) ^ k ) ) ) ) ~~> ( log ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   class class class wbr 4653   |-> cmpt 4729   o. ccom 5118   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   -oocmnf 10072   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   RR+crp 11832   (,]cioc 12176   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   *Metcxmt 19731   ballcbl 19733   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-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-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  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

This theorem is referenced by:  stirlinglem5  40295