Theorem dvlog 22760

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 2467	. . . 4 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
2	1	cnfldtop 21026	. . . . . 6 |- ( TopOpen ` ℂfld ) e. Top
3	1	cnfldtopon 21025	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
4	3	toponunii 19200	. . . . . . 7 |- CC = U. ( TopOpen ` ℂfld )
5	4	restid 14685	. . . . . 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 2480	. . . 4 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
8		cnelprrecn 9581	. . . . 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 22758	. . . . 5 |- D e. ( TopOpen ` ℂfld )
12	11	a1i 11	. . . 4 |- ( T. -> D e. ( TopOpen ` ℂfld ) )
13		logf1o 22680	. . . . . . . . 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 22751	. . . . . . . 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 672	. . . . . . 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 22672	. . . . . . . . . . 11 |- log = `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
22	21	reseq1i 5267	. . . . . . . . . 10 |- ( log |` D ) = ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D )
23	22	cnveqi 5175	. . . . . . . . 9 |- `' ( log |` D ) = `' ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D )
24		eff 13675	. . . . . . . . . . 11 |- exp : CC --> CC
25		cnvimass 5355	. . . . . . . . . . . 12 |- ( `' Im " ( -u pi (,] pi ) ) C_ dom Im
26		imf 12905	. . . . . . . . . . . . 13 |- Im : CC --> RR
27	26	fdmi 5734	. . . . . . . . . . . 12 |- dom Im = CC
28	25, 27	sseqtri 3536	. . . . . . . . . . 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 672	. . . . . . . . . 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 5657	. . . . . . . . . 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 5355	. . . . . . . . . . 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 3536	. . . . . . . . . 10 |- ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ ( `' Im " ( -u pi (,] pi ) )
37		resabs1 5300	. . . . . . . . . 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 2500	. . . . . . . 8 |- `' ( log |` D ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
40	21	imaeq1i 5332	. . . . . . . . 9 |- ( log " D ) = ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D )
41	40	reseq2i 5268	. . . . . . . 8 |- ( exp |` ( log " D ) ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
42	39, 41	eqtr4i 2499	. . . . . . 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 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 5175	. . . . . 6 |- `' `' ( log |` D ) = `' ( exp |` ( log " D ) )
48		relres 5299	. . . . . . 7 |- Rel ( log |` D )
49		dfrel2 5455	. . . . . . 7 |- ( Rel ( log |` D ) <-> `' `' ( log |` D ) = ( log |` D ) )
50	48, 49	mpbi 208	. . . . . 6 |- `' `' ( log |` D ) = ( log |` D )
51	47, 50	eqtr3i 2498	. . . . 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 12	. . . . . 6 |- ( T. -> ( log |` D ) : D --> ( log " D ) )
54		imassrn 5346	. . . . . . . 8 |- ( log " D ) C_ ran log
55		logrncn 22678	. . . . . . . . 9 |- ( x e. ran log -> x e. CC )
56	55	ssriv 3508	. . . . . . . 8 |- ran log C_ CC
57	54, 56	sstri 3513	. . . . . . 7 |- ( log " D ) C_ CC
58	10	logcn 22756	. . . . . . 7 |- ( log |` D ) e. ( D -cn-> CC )
59		cncffvrn 21137	. . . . . . 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 2559	. . . 4 |- ( T. -> `' ( exp |` ( log " D ) ) e. ( D -cn-> ( log " D ) ) )
63		ssid 3523	. . . . . . . . 9 |- CC C_ CC
64	1, 7	dvres 22050	. . . . . . . . 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 22116	. . . . . . . . 9 |- ( CC _D exp ) = exp
67	10	dvloglem 22757	. . . . . . . . . 10 |- ( log " D ) e. ( TopOpen ` ℂfld )
68		isopn3i 19349	. . . . . . . . . 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 5269	. . . . . . . 8 |- ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) = ( exp |` ( log " D ) )
71	65, 70	eqtri 2496	. . . . . . 7 |- ( CC _D ( exp |` ( log " D ) ) ) = ( exp |` ( log " D ) )
72	71	dmeqi 5202	. . . . . 6 |- dom ( CC _D ( exp |` ( log " D ) ) ) = dom ( exp |` ( log " D ) )
73		dmres 5292	. . . . . 6 |- dom ( exp |` ( log " D ) ) = ( ( log " D ) i^i dom exp )
74	24	fdmi 5734	. . . . . . . 8 |- dom exp = CC
75	57, 74	sseqtr4i 3537	. . . . . . 7 |- ( log " D ) C_ dom exp
76		df-ss 3490	. . . . . . 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 2500	. . . . 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 2671	. . . . . 6 |- -. 0 =/= 0
81		resss 5295	. . . . . . . . . . . . 13 |- ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) C_ ( CC _D exp )
82	65, 81	eqsstri 3534	. . . . . . . . . . . 12 |- ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC _D exp )
83	82, 66	sseqtri 3536	. . . . . . . . . . 11 |- ( CC _D ( exp |` ( log " D ) ) ) C_ exp
84		rnss 5229	. . . . . . . . . . 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 13691	. . . . . . . . . . 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 3513	. . . . . . . . 9 |- ran ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC \ { 0 } )
90	89	sseli 3500	. . . . . . . 8 |- ( 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) -> 0 e. ( CC \ { 0 } ) )
91		eldifsn 4152	. . . . . . . 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 22113	. . 3 |- ( T. -> ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) )
97	96	trud 1388	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) )
98	51	oveq2i 6293	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( CC _D ( log |` D ) )
99	71	fveq1i 5865	. . . . 5 |- ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = ( ( exp |` ( log " D ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) )
100		f1ocnvfv2 6169	. . . . . 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 2520	. . . 4 |- ( x e. D -> ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = x )
103	102	oveq2d 6298	. . 3 |- ( x e. D -> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) = ( 1 / x ) )
104	103	mpteq2ia 4529	. 2 |- ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) = ( x e. D |-> ( 1 / x ) )
105	97, 98, 104	3eqtr3i 2504	1 |- ( CC _D ( log |` D ) ) = ( x e. D |-> ( 1 / x ) )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   = wceq 1379   T. wtru 1380   e. wcel 1767   =/= wne 2662   \ cdif 3473   i^i cin 3475   C_ wss 3476   {csn 4027   {cpr 4029   |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   |` cres 5001   "cima 5002   Rel wrel 5004   Fun wfun 5580   -->wf 5582   -1-1->wf1 5583   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   -oocmnf 9622   -ucneg 9802   / cdiv 10202   (,]cioc 11526   Imcim 12890   expce 13655   picpi 13660   ↾_t crest 14672   TopOpenctopn 14673  ℂ_fldccnfld 18191   Topctop 19161   intcnt 19284   -cn->ccncf 21115   _D cdv 22002   logclog 22670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-tan 13665  df-pi 13666  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-cmp 19653  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006  df-log 22672

This theorem is referenced by:  dvlog2  22762  dvatan  22994  lgamgulmlem2  28212  dvcncxp1  29677  dvasin  29680