Theorem dvlog 22071

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 2438	. . . 4 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
2	1	cnfldtop 20338	. . . . . 6 |- ( TopOpen ` ℂfld ) e. Top
3	1	cnfldtopon 20337	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
4	3	toponunii 18512	. . . . . . 7 |- CC = U. ( TopOpen ` ℂfld )
5	4	restid 14364	. . . . . 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 2442	. . . 4 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
8		cnelprrecn 9367	. . . . 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 22069	. . . . 5 |- D e. ( TopOpen ` ℂfld )
12	11	a1i 11	. . . 4 |- ( T. -> D e. ( TopOpen ` ℂfld ) )
13		logf1o 21991	. . . . . . . . 9 |- log : ( CC \ { 0 } ) -1-1-onto-> ran log
14		f1of1 5635	. . . . . . . . 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 22062	. . . . . . . 8 |- D C_ ( CC \ { 0 } )
17		f1ores 5650	. . . . . . . 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 5648	. . . . . . 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 21983	. . . . . . . . . . 11 |- log = `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
22	21	reseq1i 5101	. . . . . . . . . 10 |- ( log |` D ) = ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D )
23	22	cnveqi 5009	. . . . . . . . 9 |- `' ( log |` D ) = `' ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D )
24		eff 13359	. . . . . . . . . . 11 |- exp : CC --> CC
25		cnvimass 5184	. . . . . . . . . . . 12 |- ( `' Im " ( -u pi (,] pi ) ) C_ dom Im
26		imf 12594	. . . . . . . . . . . . 13 |- Im : CC --> RR
27	26	fdmi 5559	. . . . . . . . . . . 12 |- dom Im = CC
28	25, 27	sseqtri 3383	. . . . . . . . . . 11 |- ( `' Im " ( -u pi (,] pi ) ) C_ CC
29		fssres 5573	. . . . . . . . . . 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 5556	. . . . . . . . . 10 |- ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC -> Fun ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) )
32		funcnvres2 5484	. . . . . . . . . 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 5184	. . . . . . . . . . 11 |- ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ dom ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
35	30	fdmi 5559	. . . . . . . . . . 11 |- dom ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) = ( `' Im " ( -u pi (,] pi ) )
36	34, 35	sseqtri 3383	. . . . . . . . . 10 |- ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ ( `' Im " ( -u pi (,] pi ) )
37		resabs1 5134	. . . . . . . . . 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 2462	. . . . . . . 8 |- `' ( log |` D ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
40	21	imaeq1i 5161	. . . . . . . . 9 |- ( log " D ) = ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D )
41	40	reseq2i 5102	. . . . . . . 8 |- ( exp |` ( log " D ) ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
42	39, 41	eqtr4i 2461	. . . . . . 7 |- `' ( log |` D ) = ( exp |` ( log " D ) )
43		f1oeq1 5627	. . . . . . 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 5009	. . . . . 6 |- `' `' ( log |` D ) = `' ( exp |` ( log " D ) )
48		relres 5133	. . . . . . 7 |- Rel ( log |` D )
49		dfrel2 5283	. . . . . . 7 |- ( Rel ( log |` D ) <-> `' `' ( log |` D ) = ( log |` D ) )
50	48, 49	mpbi 208	. . . . . 6 |- `' `' ( log |` D ) = ( log |` D )
51	47, 50	eqtr3i 2460	. . . . 5 |- `' ( exp |` ( log " D ) ) = ( log |` D )
52		f1of 5636	. . . . . . 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 5175	. . . . . . . 8 |- ( log " D ) C_ ran log
55		logrncn 21989	. . . . . . . . 9 |- ( x e. ran log -> x e. CC )
56	55	ssriv 3355	. . . . . . . 8 |- ran log C_ CC
57	54, 56	sstri 3360	. . . . . . 7 |- ( log " D ) C_ CC
58	10	logcn 22067	. . . . . . 7 |- ( log |` D ) e. ( D -cn-> CC )
59		cncffvrn 20449	. . . . . . 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 2522	. . . 4 |- ( T. -> `' ( exp |` ( log " D ) ) e. ( D -cn-> ( log " D ) ) )
63		ssid 3370	. . . . . . . . 9 |- CC C_ CC
64	1, 7	dvres 21361	. . . . . . . . 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 21427	. . . . . . . . 9 |- ( CC _D exp ) = exp
67	10	dvloglem 22068	. . . . . . . . . 10 |- ( log " D ) e. ( TopOpen ` ℂfld )
68		isopn3i 18661	. . . . . . . . . 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 5103	. . . . . . . 8 |- ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) = ( exp |` ( log " D ) )
71	65, 70	eqtri 2458	. . . . . . 7 |- ( CC _D ( exp |` ( log " D ) ) ) = ( exp |` ( log " D ) )
72	71	dmeqi 5036	. . . . . 6 |- dom ( CC _D ( exp |` ( log " D ) ) ) = dom ( exp |` ( log " D ) )
73		dmres 5126	. . . . . 6 |- dom ( exp |` ( log " D ) ) = ( ( log " D ) i^i dom exp )
74	24	fdmi 5559	. . . . . . . 8 |- dom exp = CC
75	57, 74	sseqtr4i 3384	. . . . . . 7 |- ( log " D ) C_ dom exp
76		df-ss 3337	. . . . . . 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 2462	. . . . 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 2608	. . . . . 6 |- -. 0 =/= 0
81		resss 5129	. . . . . . . . . . . . 13 |- ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) C_ ( CC _D exp )
82	65, 81	eqsstri 3381	. . . . . . . . . . . 12 |- ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC _D exp )
83	82, 66	sseqtri 3383	. . . . . . . . . . 11 |- ( CC _D ( exp |` ( log " D ) ) ) C_ exp
84		rnss 5063	. . . . . . . . . . 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 13375	. . . . . . . . . . 11 |- exp : CC --> ( CC \ { 0 } )
87		frn 5560	. . . . . . . . . . 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 3360	. . . . . . . . 9 |- ran ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC \ { 0 } )
90	89	sseli 3347	. . . . . . . 8 |- ( 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) -> 0 e. ( CC \ { 0 } ) )
91		eldifsn 3995	. . . . . . . 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 21424	. . 3 |- ( T. -> ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) )
97	96	trud 1378	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) )
98	51	oveq2i 6097	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( CC _D ( log |` D ) )
99	71	fveq1i 5687	. . . . 5 |- ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = ( ( exp |` ( log " D ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) )
100		f1ocnvfv2 5979	. . . . . 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 2482	. . . 4 |- ( x e. D -> ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = x )
103	102	oveq2d 6102	. . 3 |- ( x e. D -> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) = ( 1 / x ) )
104	103	mpteq2ia 4369	. 2 |- ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) = ( x e. D |-> ( 1 / x ) )
105	97, 98, 104	3eqtr3i 2466	1 |- ( CC _D ( log |` D ) ) = ( x e. D |-> ( 1 / x ) )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   = wceq 1369   T. wtru 1370   e. wcel 1756   =/= wne 2601   \ cdif 3320   i^i cin 3322   C_ wss 3323   {csn 3872   {cpr 3874   e. cmpt 4345   `'ccnv 4834   dom cdm 4835   ran crn 4836   |` cres 4837   "cima 4838   Rel wrel 4840   Fun wfun 5407   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   -oocmnf 9408   -ucneg 9588   / cdiv 9985   (,]cioc 11293   Imcim 12579   expce 13339   picpi 13344   ↾_t crest 14351   TopOpenctopn 14352  ℂ_fldccnfld 17793   Topctop 18473   intcnt 18596   -cn->ccncf 20427   _D cdv 21313   logclog 21981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-tan 13349  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-cmp 18965  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983

This theorem is referenced by:  dvlog2  22073  dvatan  22305  lgamgulmlem2  26968  dvcncxp1  28430  dvasin  28433