Theorem dvlog 22222

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 20488	. . . . . 6 |- ( TopOpen ` ℂfld ) e. Top
3	1	cnfldtopon 20487	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
4	3	toponunii 18662	. . . . . . 7 |- CC = U. ( TopOpen ` ℂfld )
5	4	restid 14483	. . . . . 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 2464	. . . 4 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
8		cnelprrecn 9479	. . . . 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 22220	. . . . 5 |- D e. ( TopOpen ` ℂfld )
12	11	a1i 11	. . . 4 |- ( T. -> D e. ( TopOpen ` ℂfld ) )
13		logf1o 22142	. . . . . . . . 9 |- log : ( CC \ { 0 } ) -1-1-onto-> ran log
14		f1of1 5741	. . . . . . . . 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 22213	. . . . . . . 8 |- D C_ ( CC \ { 0 } )
17		f1ores 5756	. . . . . . . 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 5754	. . . . . . 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 22134	. . . . . . . . . . 11 |- log = `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
22	21	reseq1i 5207	. . . . . . . . . 10 |- ( log |` D ) = ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D )
23	22	cnveqi 5115	. . . . . . . . 9 |- `' ( log |` D ) = `' ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) |` D )
24		eff 13478	. . . . . . . . . . 11 |- exp : CC --> CC
25		cnvimass 5290	. . . . . . . . . . . 12 |- ( `' Im " ( -u pi (,] pi ) ) C_ dom Im
26		imf 12713	. . . . . . . . . . . . 13 |- Im : CC --> RR
27	26	fdmi 5665	. . . . . . . . . . . 12 |- dom Im = CC
28	25, 27	sseqtri 3489	. . . . . . . . . . 11 |- ( `' Im " ( -u pi (,] pi ) ) C_ CC
29		fssres 5679	. . . . . . . . . . 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 5662	. . . . . . . . . 10 |- ( ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) : ( `' Im " ( -u pi (,] pi ) ) --> CC -> Fun ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) )
32		funcnvres2 5590	. . . . . . . . . 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 5290	. . . . . . . . . . 11 |- ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ dom ( exp |` ( `' Im " ( -u pi (,] pi ) ) )
35	30	fdmi 5665	. . . . . . . . . . 11 |- dom ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) = ( `' Im " ( -u pi (,] pi ) )
36	34, 35	sseqtri 3489	. . . . . . . . . 10 |- ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) C_ ( `' Im " ( -u pi (,] pi ) )
37		resabs1 5240	. . . . . . . . . 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 2484	. . . . . . . 8 |- `' ( log |` D ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
40	21	imaeq1i 5267	. . . . . . . . 9 |- ( log " D ) = ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D )
41	40	reseq2i 5208	. . . . . . . 8 |- ( exp |` ( log " D ) ) = ( exp |` ( `' ( exp |` ( `' Im " ( -u pi (,] pi ) ) ) " D ) )
42	39, 41	eqtr4i 2483	. . . . . . 7 |- `' ( log |` D ) = ( exp |` ( log " D ) )
43		f1oeq1 5733	. . . . . . 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 5115	. . . . . 6 |- `' `' ( log |` D ) = `' ( exp |` ( log " D ) )
48		relres 5239	. . . . . . 7 |- Rel ( log |` D )
49		dfrel2 5389	. . . . . . 7 |- ( Rel ( log |` D ) <-> `' `' ( log |` D ) = ( log |` D ) )
50	48, 49	mpbi 208	. . . . . 6 |- `' `' ( log |` D ) = ( log |` D )
51	47, 50	eqtr3i 2482	. . . . 5 |- `' ( exp |` ( log " D ) ) = ( log |` D )
52		f1of 5742	. . . . . . 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 5281	. . . . . . . 8 |- ( log " D ) C_ ran log
55		logrncn 22140	. . . . . . . . 9 |- ( x e. ran log -> x e. CC )
56	55	ssriv 3461	. . . . . . . 8 |- ran log C_ CC
57	54, 56	sstri 3466	. . . . . . 7 |- ( log " D ) C_ CC
58	10	logcn 22218	. . . . . . 7 |- ( log |` D ) e. ( D -cn-> CC )
59		cncffvrn 20599	. . . . . . 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 2543	. . . 4 |- ( T. -> `' ( exp |` ( log " D ) ) e. ( D -cn-> ( log " D ) ) )
63		ssid 3476	. . . . . . . . 9 |- CC C_ CC
64	1, 7	dvres 21512	. . . . . . . . 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 21578	. . . . . . . . 9 |- ( CC _D exp ) = exp
67	10	dvloglem 22219	. . . . . . . . . 10 |- ( log " D ) e. ( TopOpen ` ℂfld )
68		isopn3i 18811	. . . . . . . . . 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 5209	. . . . . . . 8 |- ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) = ( exp |` ( log " D ) )
71	65, 70	eqtri 2480	. . . . . . 7 |- ( CC _D ( exp |` ( log " D ) ) ) = ( exp |` ( log " D ) )
72	71	dmeqi 5142	. . . . . 6 |- dom ( CC _D ( exp |` ( log " D ) ) ) = dom ( exp |` ( log " D ) )
73		dmres 5232	. . . . . 6 |- dom ( exp |` ( log " D ) ) = ( ( log " D ) i^i dom exp )
74	24	fdmi 5665	. . . . . . . 8 |- dom exp = CC
75	57, 74	sseqtr4i 3490	. . . . . . 7 |- ( log " D ) C_ dom exp
76		df-ss 3443	. . . . . . 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 2484	. . . . 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 2653	. . . . . 6 |- -. 0 =/= 0
81		resss 5235	. . . . . . . . . . . . 13 |- ( ( CC _D exp ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` ( log " D ) ) ) C_ ( CC _D exp )
82	65, 81	eqsstri 3487	. . . . . . . . . . . 12 |- ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC _D exp )
83	82, 66	sseqtri 3489	. . . . . . . . . . 11 |- ( CC _D ( exp |` ( log " D ) ) ) C_ exp
84		rnss 5169	. . . . . . . . . . 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 13494	. . . . . . . . . . 11 |- exp : CC --> ( CC \ { 0 } )
87		frn 5666	. . . . . . . . . . 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 3466	. . . . . . . . 9 |- ran ( CC _D ( exp |` ( log " D ) ) ) C_ ( CC \ { 0 } )
90	89	sseli 3453	. . . . . . . 8 |- ( 0 e. ran ( CC _D ( exp |` ( log " D ) ) ) -> 0 e. ( CC \ { 0 } ) )
91		eldifsn 4101	. . . . . . . 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 21575	. . 3 |- ( T. -> ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) )
97	96	trud 1379	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) )
98	51	oveq2i 6204	. 2 |- ( CC _D `' ( exp |` ( log " D ) ) ) = ( CC _D ( log |` D ) )
99	71	fveq1i 5793	. . . . 5 |- ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = ( ( exp |` ( log " D ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) )
100		f1ocnvfv2 6086	. . . . . 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 2504	. . . 4 |- ( x e. D -> ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) = x )
103	102	oveq2d 6209	. . 3 |- ( x e. D -> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) = ( 1 / x ) )
104	103	mpteq2ia 4475	. 2 |- ( x e. D |-> ( 1 / ( ( CC _D ( exp |` ( log " D ) ) ) ` ( `' ( exp |` ( log " D ) ) ` x ) ) ) ) = ( x e. D |-> ( 1 / x ) )
105	97, 98, 104	3eqtr3i 2488	1 |- ( CC _D ( log |` D ) ) = ( x e. D |-> ( 1 / x ) )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   = wceq 1370   T. wtru 1371   e. wcel 1758   =/= wne 2644   \ cdif 3426   i^i cin 3428   C_ wss 3429   {csn 3978   {cpr 3980   |-> cmpt 4451   `'ccnv 4940   dom cdm 4941   ran crn 4942   |` cres 4943   "cima 4944   Rel wrel 4946   Fun wfun 5513   -->wf 5515   -1-1->wf1 5516   -1-1-onto->wf1o 5518   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387   -oocmnf 9520   -ucneg 9700   / cdiv 10097   (,]cioc 11405   Imcim 12698   expce 13458   picpi 13463   ↾_t crest 14470   TopOpenctopn 14471  ℂ_fldccnfld 17936   Topctop 18623   intcnt 18746   -cn->ccncf 20577   _D cdv 21464   logclog 22132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ioc 11409  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464  df-sin 13466  df-cos 13467  df-tan 13468  df-pi 13469  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-lp 18865  df-perf 18866  df-cn 18956  df-cnp 18957  df-haus 19044  df-cmp 19115  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-xms 20020  df-ms 20021  df-tms 20022  df-cncf 20579  df-limc 21467  df-dv 21468  df-log 22134

This theorem is referenced by:  dvlog2  22224  dvatan  22456  lgamgulmlem2  27153  dvcncxp1  28618  dvasin  28621