Theorem dvlog 23596

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 2451	. . . 4 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
2	1	cnfldtop 21804	. . . . . 6 |- ( TopOpen ` ℂfld ) e. Top
3	1	cnfldtopon 21803	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
4	3	toponunii 19947	. . . . . . 7 |- CC = U. ( TopOpen ` ℂfld )
5	4	restid 15332	. . . . . 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 2460	. . . 4 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
8		cnelprrecn 9632	. . . . 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 23594	. . . . 5 |- D e. ( TopOpen ` ℂfld )
12	11	a1i 11	. . . 4 |- ( T. -> D e. ( TopOpen ` ℂfld ) )
13		logf1o 23514	. . . . . . . . 9 |- log : ( CC \ { 0 } ) -1-1-onto-> ran log
14		f1of1 5813	. . . . . . . . 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 23587	. . . . . . . 8 |- D C_ ( CC \ { 0 } )
17		f1ores 5828	. . . . . . . 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 678	. . . . . . 7 |- ( log |` D ) : D -1-1-onto-> ( log " D )
19		f1ocnv 5826	. . . . . . 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 23506	. . . . . . . . . . 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 14136	. . . . . . . . . . 11 |- exp : CC --> CC
25		cnvimass 5188	. . . . . . . . . . . 12 |- ( `' Im " ( -u pi (,] pi ) ) C_ dom Im
26		imf 13176	. . . . . . . . . . . . 13 |- Im : CC --> RR
27	26	fdmi 5734	. . . . . . . . . . . 12 |- dom Im = CC
28	25, 27	sseqtri 3464	. . . . . . . . . . 11 |- ( `' Im " ( -u pi (,] pi ) ) C_ CC
29		fssres 5749	. . . . . . . . . . 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 678	. . . . . . . . . 10 |- ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC
31		ffun 5731	. . . . . . . . . 10 |- ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC -> Fun ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) )
32		funcnvres2 5654	. . . . . . . . . 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 5188	. . . . . . . . . . 11 |- ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ dom ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
35	30	fdmi 5734	. . . . . . . . . . 11 |- dom ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) = ( `' Im " ( -u pi (,] pi ) )
36	34, 35	sseqtri 3464	. . . . . . . . . 10 |- ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ ( `' Im " ( -u pi (,] pi ) )
37		resabs1 5133	. . . . . . . . . 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 2477	. . . . . . . 8 |- `' ( log |` D ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
40	21	imaeq1i 5165	. . . . . . . . 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 2476	. . . . . . 7 |- `' ( log |` D ) = ( exp |` ( log " D ) )
43		f1oeq1 5805	. . . . . . 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 212	. . . . 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 5132	. . . . . . 7 |- Rel ( log |` D )
49		dfrel2 5286	. . . . . . 7 |- ( Rel ( log |` D ) <-> `' `' ( log |` D ) = ( log |` D ) )
50	48, 49	mpbi 212	. . . . . 6 |- `' `' ( log |` D ) = ( log |` D )
51	47, 50	eqtr3i 2475	. . . . 5 |- `' ( exp |` ( log " D ) ) = ( log |` D )
52		f1of 5814	. . . . . . 7 |- ( ( log |` D ) : D -1-1-onto-> ( log " D ) -> ( log |` D ) : D --> ( log " D ) )
53	18, 52	mp1i 13	. . . . . 6 |- ( T. -> ( log |` D ) : D --> ( log " D ) )
54		imassrn 5179	. . . . . . . 8 |- ( log " D ) C_ ran log
55		logrncn 23512	. . . . . . . . 9 |- ( x e. ran log -> x e. CC )
56	55	ssriv 3436	. . . . . . . 8 |- ran log C_ CC
57	54, 56	sstri 3441	. . . . . . 7 |- ( log " D ) C_ CC
58	10	logcn 23592	. . . . . . 7 |- ( log |` D ) e. ( D -cn-> CC )
59		cncffvrn 21930	. . . . . . 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 678	. . . . . 6 |- ( ( log |` D ) e. ( D -cn-> ( log " D ) ) <-> ( log |` D ) : D --> ( log " D ) )
61	53, 60	sylibr 216	. . . . 5 |- ( T. -> ( log |` D ) e. ( D -cn-> ( log " D ) ) )
62	51, 61	syl5eqel 2533	. . . 4 |- ( T. -> `' ( exp |` ( log " D ) ) e. ( D -cn-> ( log " D ) ) )
63		ssid 3451	. . . . . . . . 9 |- CC C_ CC
64	1, 7	dvres 22866	. . . . . . . . 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 679	. . . . . . . 8 |- ( CC _D ( exp |` ( log " D ) ) ) = ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) )
66		dvef 22932	. . . . . . . . 9 |- ( CC _D exp ) = exp
67	10	dvloglem 23593	. . . . . . . . . 10 |- ( log " D ) e. ( TopOpen ` ℂfld )
68		isopn3i 20098	. . . . . . . . . 10 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ ( log " D ) e. ( TopOpen ` ℂfld ) ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) = ( log " D ) )
69	2, 67, 68	mp2an 678	. . . . . . . . 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 2473	. . . . . . 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 5125	. . . . . 6 |- dom ( exp |` ( log " D ) ) = ( ( log " D ) i^i dom exp )
74	24	fdmi 5734	. . . . . . . 8 |- dom exp = CC
75	57, 74	sseqtr4i 3465	. . . . . . 7 |- ( log " D ) C_ dom exp
76		df-ss 3418	. . . . . . 7 |- ( ( log " D ) C_ dom exp <-> ( ( log " D ) i^i dom exp ) = ( log " D ) )
77	75, 76	mpbi 212	. . . . . 6 |- ( ( log " D ) i^i dom exp ) = ( log " D )
78	72, 73, 77	3eqtri 2477	. . . . 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 2632	. . . . . 6 |- -. 0 =/= 0
81		resss 5128	. . . . . . . . . . . . 13 |- ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) C_ ( CC _D exp )
82	65, 81	eqsstri 3462	. . . . . . . . . . . 12 |- ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC _D exp )
83	82, 66	sseqtri 3464	. . . . . . . . . . 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 14153	. . . . . . . . . . 11 |- exp : CC --> ( CC \ { 0 } )
87		frn 5735	. . . . . . . . . . 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 3441	. . . . . . . . 9 |- ran ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC \ { 0 } )
90	89	sseli 3428	. . . . . . . 8 |- ( 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) -> 0 e. ( CC \ { 0 } ) )
91		eldifsn 4097	. . . . . . . 8 |- ( 0 e. ( CC \ { 0 } ) <-> ( 0 e. CC /\ 0 =/= 0 ) )
92	90, 91	sylib 200	. . . . . . 7 |- ( 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) -> ( 0 e. CC /\ 0 =/= 0 ) )
93	92	simprd 465	. . . . . 6 |- ( 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) -> 0 =/= 0 )
94	80, 93	mto 180	. . . . 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 22929	. . 3 |- ( T. -> ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) )
97	96	trud 1453	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) )
98	51	oveq2i 6301	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( CC _D ( log |` D ) )
99	71	fveq1i 5866	. . . . 5 |- ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = ( ( exp |` ( log " D ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) )
100		f1ocnvfv2 6176	. . . . . 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 676	. . . . 5 |- ( x e. D -> ( ( exp |` ( log " D ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = x )
102	99, 101	syl5eq 2497	. . . 4 |- ( x e. D -> ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = x )
103	102	oveq2d 6306	. . 3 |- ( x e. D -> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) = ( 1 / x ) )
104	103	mpteq2ia 4485	. 2 |- ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) = ( x e. D |-> ( 1 / x ) )
105	97, 98, 104	3eqtr3i 2481	1 |- ( CC _D ( log |` D ) ) = ( x e. D |-> ( 1 / x ) )

Syntax hints:   -. wn 3   <-> wb 188   /\ wa 371   = wceq 1444   T. wtru 1445   e. wcel 1887   =/= wne 2622   \ cdif 3401   i^i cin 3403   C_ wss 3404   {csn 3968   {cpr 3970   |-> cmpt 4461   `'ccnv 4833   dom cdm 4834   ran crn 4835   |` cres 4836   "cima 4837   Rel wrel 4839   Fun wfun 5576   -->wf 5578   -1-1->wf1 5579   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   -oocmnf 9673   -ucneg 9861   / cdiv 10269   (,]cioc 11636   Imcim 13161   expce 14114   picpi 14119   ↾_t crest 15319   TopOpenctopn 15320  ℂ_fldccnfld 18970   Topctop 19917   intcnt 20032   -cn->ccncf 21908   _D cdv 22818   logclog 23504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-tan 14125  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506

This theorem is referenced by:  dvlog2  23598  dvcncxp1  23683  dvatan  23861  lgamgulmlem2  23955  dvasin  32028