Theorem dvlog 23157

Description: The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)

Hypothesis

Ref	Expression
logcn.d	|- D = ( CC \ ( -oo (,] 0 ) )

Assertion

Ref	Expression
dvlog	|- ( CC _D ( log |` D ) ) = ( x e. D |-> ( 1 / x ) )

Distinct variable group:   x, D

Proof of Theorem dvlog

Step	Hyp	Ref	Expression
1		eqid 2457	. . . 4 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
2	1	cnfldtop 21416	. . . . . 6 |- ( TopOpen ` ℂfld ) e. Top
3	1	cnfldtopon 21415	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
4	3	toponunii 19559	. . . . . . 7 |- CC = U. ( TopOpen ` ℂfld )
5	4	restid 14850	. . . . . 6 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
6	2, 5	ax-mp 5	. . . . 5 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
7	6	eqcomi 2470	. . . 4 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
8		cnelprrecn 9602	. . . . 5 |- CC e. { RR , CC }
9	8	a1i 11	. . . 4 |- ( T. -> CC e. { RR , CC } )
10		logcn.d	. . . . . 6 |- D = ( CC \ ( -oo (,] 0 ) )
11	10	logdmopn 23155	. . . . 5 |- D e. ( TopOpen ` ℂfld )
12	11	a1i 11	. . . 4 |- ( T. -> D e. ( TopOpen ` ℂfld ) )
13		logf1o 23077	. . . . . . . . 9 |- log : ( CC \ { 0 } ) -1-1-onto-> ran log
14		f1of1 5821	. . . . . . . . 9 |- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> log : ( CC \ { 0 } ) -1-1-> ran log )
15	13, 14	ax-mp 5	. . . . . . . 8 |- log : ( CC \ { 0 } ) -1-1-> ran log
16	10	logdmss 23148	. . . . . . . 8 |- D C_ ( CC \ { 0 } )
17		f1ores 5836	. . . . . . . 8 |- ( ( log : ( CC \ { 0 } ) -1-1-> ran log /\ D C_ ( CC \ { 0 } ) ) -> ( log |` D ) : D -1-1-onto-> ( log " D ) )
18	15, 16, 17	mp2an 672	. . . . . . 7 |- ( log |` D ) : D -1-1-onto-> ( log " D )
19		f1ocnv 5834	. . . . . . 7 |- ( ( log |` D ) : D -1-1-onto-> ( log " D ) -> `' ( log |` D ) : ( log " D ) -1-1-onto-> D )
20	18, 19	ax-mp 5	. . . . . 6 |- `' ( log |` D ) : ( log " D ) -1-1-onto-> D
21		df-log 23069	. . . . . . . . . . 11 |- log = `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
22	21	reseq1i 5279	. . . . . . . . . 10 |- ( log |` D ) = ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D )
23	22	cnveqi 5187	. . . . . . . . 9 |- `' ( log |` D ) = `' ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D )
24		eff 13828	. . . . . . . . . . 11 |- exp : CC --> CC
25		cnvimass 5367	. . . . . . . . . . . 12 |- ( `' Im " ( -u pi (,] pi ) ) C_ dom Im
26		imf 12957	. . . . . . . . . . . . 13 |- Im : CC --> RR
27	26	fdmi 5742	. . . . . . . . . . . 12 |- dom Im = CC
28	25, 27	sseqtri 3531	. . . . . . . . . . 11 |- ( `' Im " ( -u pi (,] pi ) ) C_ CC
29		fssres 5757	. . . . . . . . . . 11 |- ( ( exp : CC --> CC /\ ( `' Im " ( -u pi (,] pi ) ) C_ CC ) -> ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC )
30	24, 28, 29	mp2an 672	. . . . . . . . . 10 |- ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC
31		ffun 5739	. . . . . . . . . 10 |- ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC -> Fun ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) )
32		funcnvres2 5665	. . . . . . . . . 10 |- ( Fun ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) -> `' ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D ) = ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) ) )
33	30, 31, 32	mp2b 10	. . . . . . . . 9 |- `' ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D ) = ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
34		cnvimass 5367	. . . . . . . . . . 11 |- ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ dom ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
35	30	fdmi 5742	. . . . . . . . . . 11 |- dom ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) = ( `' Im " ( -u pi (,] pi ) )
36	34, 35	sseqtri 3531	. . . . . . . . . 10 |- ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ ( `' Im " ( -u pi (,] pi ) )
37		resabs1 5312	. . . . . . . . . 10 |- ( ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ ( `' Im " ( -u pi (,] pi ) ) -> ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) ) )
38	36, 37	ax-mp 5	. . . . . . . . 9 |- ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
39	23, 33, 38	3eqtri 2490	. . . . . . . 8 |- `' ( log |` D ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
40	21	imaeq1i 5344	. . . . . . . . 9 |- ( log " D ) = ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D )
41	40	reseq2i 5280	. . . . . . . 8 |- ( exp |` ( log " D ) ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
42	39, 41	eqtr4i 2489	. . . . . . 7 |- `' ( log |` D ) = ( exp |` ( log " D ) )
43		f1oeq1 5813	. . . . . . 7 |- ( `' ( log |` D ) = ( exp |` ( log " D ) ) -> ( `' ( log |` D ) : ( log " D ) -1-1-onto-> D <-> ( exp |` ( log " D ) ) : ( log " D ) -1-1-onto-> D ) )
44	42, 43	ax-mp 5	. . . . . 6 |- ( `' ( log |` D ) : ( log " D ) -1-1-onto-> D <-> ( exp |` ( log " D ) ) : ( log " D ) -1-1-onto-> D )
45	20, 44	mpbi 208	. . . . 5 |- ( exp |` ( log " D ) ) : ( log " D ) -1-1-onto-> D
46	45	a1i 11	. . . 4 |- ( T. -> ( exp |` ( log " D ) ) : ( log " D ) -1-1-onto-> D )
47	42	cnveqi 5187	. . . . . 6 |- `' `' ( log |` D ) = `' ( exp |` ( log " D ) )
48		relres 5311	. . . . . . 7 |- Rel ( log |` D )
49		dfrel2 5463	. . . . . . 7 |- ( Rel ( log |` D ) <-> `' `' ( log |` D ) = ( log |` D ) )
50	48, 49	mpbi 208	. . . . . 6 |- `' `' ( log |` D ) = ( log |` D )
51	47, 50	eqtr3i 2488	. . . . 5 |- `' ( exp |` ( log " D ) ) = ( log |` D )
52		f1of 5822	. . . . . . 7 |- ( ( log |` D ) : D -1-1-onto-> ( log " D ) -> ( log |` D ) : D --> ( log " D ) )
53	18, 52	mp1i 12	. . . . . 6 |- ( T. -> ( log |` D ) : D --> ( log " D ) )
54		imassrn 5358	. . . . . . . 8 |- ( log " D ) C_ ran log
55		logrncn 23075	. . . . . . . . 9 |- ( x e. ran log -> x e. CC )
56	55	ssriv 3503	. . . . . . . 8 |- ran log C_ CC
57	54, 56	sstri 3508	. . . . . . 7 |- ( log " D ) C_ CC
58	10	logcn 23153	. . . . . . 7 |- ( log |` D ) e. ( D -cn-> CC )
59		cncffvrn 21527	. . . . . . 7 |- ( ( ( log " D ) C_ CC /\ ( log |` D ) e. ( D -cn-> CC ) ) -> ( ( log |` D ) e. ( D -cn-> ( log " D ) ) <-> ( log |` D ) : D --> ( log " D ) ) )
60	57, 58, 59	mp2an 672	. . . . . 6 |- ( ( log |` D ) e. ( D -cn-> ( log " D ) ) <-> ( log |` D ) : D --> ( log " D ) )
61	53, 60	sylibr 212	. . . . 5 |- ( T. -> ( log |` D ) e. ( D -cn-> ( log " D ) ) )
62	51, 61	syl5eqel 2549	. . . 4 |- ( T. -> `' ( exp |` ( log " D ) ) e. ( D -cn-> ( log " D ) ) )
63		ssid 3518	. . . . . . . . 9 |- CC C_ CC
64	1, 7	dvres 22440	. . . . . . . . 9 |- ( ( ( CC C_ CC /\ exp : CC --> CC ) /\ ( CC C_ CC /\ ( log " D ) C_ CC ) ) -> ( CC _D ( exp |` ( log " D ) ) ) = ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) )
65	63, 24, 63, 57, 64	mp4an 673	. . . . . . . 8 |- ( CC _D ( exp |` ( log " D ) ) ) = ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) )
66		dvef 22506	. . . . . . . . 9 |- ( CC _D exp ) = exp
67	10	dvloglem 23154	. . . . . . . . . 10 |- ( log " D ) e. ( TopOpen ` ℂfld )
68		isopn3i 19709	. . . . . . . . . 10 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ ( log " D ) e. ( TopOpen ` ℂfld ) ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) = ( log " D ) )
69	2, 67, 68	mp2an 672	. . . . . . . . 9 |- ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) = ( log " D )
70	66, 69	reseq12i 5281	. . . . . . . 8 |- ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) = ( exp |` ( log " D ) )
71	65, 70	eqtri 2486	. . . . . . 7 |- ( CC _D ( exp |` ( log " D ) ) ) = ( exp |` ( log " D ) )
72	71	dmeqi 5214	. . . . . 6 |- dom ( CC _D ( exp |` ( log " D ) ) ) = dom ( exp |` ( log " D ) )
73		dmres 5304	. . . . . 6 |- dom ( exp |` ( log " D ) ) = ( ( log " D ) i^i dom exp )
74	24	fdmi 5742	. . . . . . . 8 |- dom exp = CC
75	57, 74	sseqtr4i 3532	. . . . . . 7 |- ( log " D ) C_ dom exp
76		df-ss 3485	. . . . . . 7 |- ( ( log " D ) C_ dom exp <-> ( ( log " D ) i^i dom exp ) = ( log " D ) )
77	75, 76	mpbi 208	. . . . . 6 |- ( ( log " D ) i^i dom exp ) = ( log " D )
78	72, 73, 77	3eqtri 2490	. . . . 5 |- dom ( CC _D ( exp |` ( log " D ) ) ) = ( log " D )
79	78	a1i 11	. . . 4 |- ( T. -> dom ( CC _D ( exp |` ( log " D ) ) ) = ( log " D ) )
80		neirr 2661	. . . . . 6 |- -. 0 =/= 0
81		resss 5307	. . . . . . . . . . . . 13 |- ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) C_ ( CC _D exp )
82	65, 81	eqsstri 3529	. . . . . . . . . . . 12 |- ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC _D exp )
83	82, 66	sseqtri 3531	. . . . . . . . . . 11 |- ( CC _D ( exp |` ( log " D ) ) ) C_ exp
84		rnss 5241	. . . . . . . . . . 11 |- ( ( CC _D ( exp |` ( log " D ) ) ) C_ exp -> ran ( CC _D ( exp |` ( log " D ) ) ) C_ ran exp )
85	83, 84	ax-mp 5	. . . . . . . . . 10 |- ran ( CC _D ( exp |` ( log " D ) ) ) C_ ran exp
86		eff2 13845	. . . . . . . . . . 11 |- exp : CC --> ( CC \ { 0 } )
87		frn 5743	. . . . . . . . . . 11 |- ( exp : CC --> ( CC \ { 0 } ) -> ran exp C_ ( CC \ { 0 } ) )
88	86, 87	ax-mp 5	. . . . . . . . . 10 |- ran exp C_ ( CC \ { 0 } )
89	85, 88	sstri 3508	. . . . . . . . 9 |- ran ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC \ { 0 } )
90	89	sseli 3495	. . . . . . . 8 |- ( 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) -> 0 e. ( CC \ { 0 } ) )
91		eldifsn 4157	. . . . . . . 8 |- ( 0 e. ( CC \ { 0 } ) <-> ( 0 e. CC /\ 0 =/= 0 ) )
92	90, 91	sylib 196	. . . . . . 7 |- ( 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) -> ( 0 e. CC /\ 0 =/= 0 ) )
93	92	simprd 463	. . . . . 6 |- ( 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) -> 0 =/= 0 )
94	80, 93	mto 176	. . . . 5 |- -. 0 e. ran ( CC _D ( exp |` ( log " D ) ) )
95	94	a1i 11	. . . 4 |- ( T. -> -. 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) )
96	1, 7, 9, 12, 46, 62, 79, 95	dvcnv 22503	. . 3 |- ( T. -> ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) )
97	96	trud 1404	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) )
98	51	oveq2i 6307	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( CC _D ( log |` D ) )
99	71	fveq1i 5873	. . . . 5 |- ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = ( ( exp |` ( log " D ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) )
100		f1ocnvfv2 6184	. . . . . 6 |- ( ( ( exp |` ( log " D ) ) : ( log " D ) -1-1-onto-> D /\ x e. D ) -> ( ( exp |` ( log " D ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = x )
101	45, 100	mpan 670	. . . . 5 |- ( x e. D -> ( ( exp |` ( log " D ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = x )
102	99, 101	syl5eq 2510	. . . 4 |- ( x e. D -> ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = x )
103	102	oveq2d 6312	. . 3 |- ( x e. D -> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) = ( 1 / x ) )
104	103	mpteq2ia 4539	. 2 |- ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) = ( x e. D |-> ( 1 / x ) )
105	97, 98, 104	3eqtr3i 2494	1 |- ( CC _D ( log |` D ) ) = ( x e. D |-> ( 1 / x ) )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   = wceq 1395   T. wtru 1396   e. wcel 1819   =/= wne 2652   \ cdif 3468   i^i cin 3470   C_ wss 3471   {csn 4032   {cpr 4034   |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   ran crn 5009   |` cres 5010   "cima 5011   Rel wrel 5013   Fun wfun 5588   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   -oocmnf 9643   -ucneg 9825   / cdiv 10227   (,]cioc 11555   Imcim 12942   expce 13808   picpi 13813   ↾_t crest 14837   TopOpenctopn 14838  ℂ_fldccnfld 18546   Topctop 19520   intcnt 19644   -cn->ccncf 21505   _D cdv 22392   logclog 23067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-shft 12911  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-limsup 13305  df-clim 13322  df-rlim 13323  df-sum 13520  df-ef 13814  df-sin 13816  df-cos 13817  df-tan 13818  df-pi 13819  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lp 19763  df-perf 19764  df-cn 19854  df-cnp 19855  df-haus 19942  df-cmp 20013  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507  df-limc 22395  df-dv 22396  df-log 23069

This theorem is referenced by:  dvlog2  23159  dvatan  23391  lgamgulmlem2  28747  dvcncxp1  30262  dvasin  30265