Theorem dvcjbr 23712

Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 23713. (This doesn't follow from dvcobr 23709 because * is not a function on the reals, and even if we used complex derivatives, * is not complex-differentiable.) (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Hypotheses

Ref	Expression
dvcj.f	|- ( ph -> F : X --> CC )
dvcj.x	|- ( ph -> X C_ RR )
dvcj.c	|- ( ph -> C e. dom ( RR _D F ) )

Assertion

Ref	Expression
dvcjbr	|- ( ph -> C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) )

Proof of Theorem dvcjbr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-resscn 9993	. . . . 5 |- RR C_ CC
2	1	a1i 11	. . . 4 |- ( ph -> RR C_ CC )
3		dvcj.f	. . . 4 |- ( ph -> F : X --> CC )
4		dvcj.x	. . . 4 |- ( ph -> X C_ RR )
5		eqid 2622	. . . . 5 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
6	5	tgioo2 22606	. . . 4 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
7	2, 3, 4, 6, 5	dvbssntr 23664	. . 3 |- ( ph -> dom ( RR _D F ) C_ ( ( int ` ( topGen ` ran (,) ) ) ` X ) )
8		dvcj.c	. . 3 |- ( ph -> C e. dom ( RR _D F ) )
9	7, 8	sseldd 3604	. 2 |- ( ph -> C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) )
10	4, 1	syl6ss 3615	. . . . . 6 |- ( ph -> X C_ CC )
11	1	a1i 11	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> RR C_ CC )
12		simpl 473	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> F : X --> CC )
13		simpr 477	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> X C_ RR )
14	11, 12, 13	dvbss 23665	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D F ) C_ X )
15	3, 4, 14	syl2anc 693	. . . . . . 7 |- ( ph -> dom ( RR _D F ) C_ X )
16	15, 8	sseldd 3604	. . . . . 6 |- ( ph -> C e. X )
17	3, 10, 16	dvlem 23660	. . . . 5 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) e. CC )
18		eqid 2622	. . . . 5 |- ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) )
19	17, 18	fmptd 6385	. . . 4 |- ( ph -> ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) : ( X \ { C } ) --> CC )
20		ssid 3624	. . . . 5 |- CC C_ CC
21	20	a1i 11	. . . 4 |- ( ph -> CC C_ CC )
22	5	cnfldtopon 22586	. . . . . 6 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
23	22	toponunii 20721	. . . . . . 7 |- CC = U. ( TopOpen ` ℂfld )
24	23	restid 16094	. . . . . 6 |- ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
25	22, 24	ax-mp 5	. . . . 5 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
26	25	eqcomi 2631	. . . 4 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
27		dvf 23671	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
28		ffun 6048	. . . . . . . 8 |- ( ( RR _D F ) : dom ( RR _D F ) --> CC -> Fun ( RR _D F ) )
29		funfvbrb 6330	. . . . . . . 8 |- ( Fun ( RR _D F ) -> ( C e. dom ( RR _D F ) <-> C ( RR _D F ) ( ( RR _D F ) ` C ) ) )
30	27, 28, 29	mp2b 10	. . . . . . 7 |- ( C e. dom ( RR _D F ) <-> C ( RR _D F ) ( ( RR _D F ) ` C ) )
31	8, 30	sylib 208	. . . . . 6 |- ( ph -> C ( RR _D F ) ( ( RR _D F ) ` C ) )
32	6, 5, 18, 2, 3, 4	eldv 23662	. . . . . 6 |- ( ph -> ( C ( RR _D F ) ( ( RR _D F ) ` C ) <-> ( C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) /\ ( ( RR _D F ) ` C ) e. ( ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) lim CC C ) ) ) )
33	31, 32	mpbid 222	. . . . 5 |- ( ph -> ( C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) /\ ( ( RR _D F ) ` C ) e. ( ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) lim CC C ) ) )
34	33	simprd 479	. . . 4 |- ( ph -> ( ( RR _D F ) ` C ) e. ( ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) lim CC C ) )
35		cjcncf 22707	. . . . . 6 |- * e. ( CC -cn-> CC )
36	5	cncfcn1 22713	. . . . . 6 |- ( CC -cn-> CC ) = ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) )
37	35, 36	eleqtri 2699	. . . . 5 |- * e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) )
38	27	ffvelrni 6358	. . . . . 6 |- ( C e. dom ( RR _D F ) -> ( ( RR _D F ) ` C ) e. CC )
39	8, 38	syl 17	. . . . 5 |- ( ph -> ( ( RR _D F ) ` C ) e. CC )
40	23	cncnpi 21082	. . . . 5 |- ( ( * e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) /\ ( ( RR _D F ) ` C ) e. CC ) -> * e. ( ( ( TopOpen ` ℂfld ) CnP ( TopOpen ` ℂfld ) ) ` ( ( RR _D F ) ` C ) ) )
41	37, 39, 40	sylancr 695	. . . 4 |- ( ph -> * e. ( ( ( TopOpen ` ℂfld ) CnP ( TopOpen ` ℂfld ) ) ` ( ( RR _D F ) ` C ) ) )
42	19, 21, 5, 26, 34, 41	limccnp 23655	. . 3 |- ( ph -> ( * ` ( ( RR _D F ) ` C ) ) e. ( ( * o. ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) lim CC C ) )
43		eqidd 2623	. . . . . 6 |- ( ph -> ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) )
44		cjf 13844	. . . . . . . 8 |- * : CC --> CC
45	44	a1i 11	. . . . . . 7 |- ( ph -> * : CC --> CC )
46	45	feqmptd 6249	. . . . . 6 |- ( ph -> * = ( y e. CC |-> ( * ` y ) ) )
47		fveq2 6191	. . . . . 6 |- ( y = ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) -> ( * ` y ) = ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) )
48	17, 43, 46, 47	fmptco 6396	. . . . 5 |- ( ph -> ( * o. ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) = ( x e. ( X \ { C } ) |-> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) )
49	3	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( X \ { C } ) ) -> F : X --> CC )
50		eldifi 3732	. . . . . . . . . . 11 |- ( x e. ( X \ { C } ) -> x e. X )
51	50	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ x e. ( X \ { C } ) ) -> x e. X )
52	49, 51	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( F ` x ) e. CC )
53	3, 16	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( F ` C ) e. CC )
54	53	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( F ` C ) e. CC )
55	52, 54	subcld 10392	. . . . . . . 8 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( ( F ` x ) - ( F ` C ) ) e. CC )
56	4	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> x e. RR )
57	50, 56	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ x e. ( X \ { C } ) ) -> x e. RR )
58	4, 16	sseldd 3604	. . . . . . . . . . 11 |- ( ph -> C e. RR )
59	58	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( X \ { C } ) ) -> C e. RR )
60	57, 59	resubcld 10458	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( x - C ) e. RR )
61	60	recnd 10068	. . . . . . . 8 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( x - C ) e. CC )
62	57	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { C } ) ) -> x e. CC )
63	59	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { C } ) ) -> C e. CC )
64		eldifsni 4320	. . . . . . . . . 10 |- ( x e. ( X \ { C } ) -> x =/= C )
65	64	adantl 482	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { C } ) ) -> x =/= C )
66	62, 63, 65	subne0d 10401	. . . . . . . 8 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( x - C ) =/= 0 )
67	55, 61, 66	cjdivd 13963	. . . . . . 7 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( ( * ` ( ( F ` x ) - ( F ` C ) ) ) / ( * ` ( x - C ) ) ) )
68		cjsub 13889	. . . . . . . . . 10 |- ( ( ( F ` x ) e. CC /\ ( F ` C ) e. CC ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
69	52, 54, 68	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
70		fvco3 6275	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ x e. X ) -> ( ( * o. F ) ` x ) = ( * ` ( F ` x ) ) )
71	3, 50, 70	syl2an 494	. . . . . . . . . 10 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( ( * o. F ) ` x ) = ( * ` ( F ` x ) ) )
72		fvco3 6275	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ C e. X ) -> ( ( * o. F ) ` C ) = ( * ` ( F ` C ) ) )
73	3, 16, 72	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( ( * o. F ) ` C ) = ( * ` ( F ` C ) ) )
74	73	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( ( * o. F ) ` C ) = ( * ` ( F ` C ) ) )
75	71, 74	oveq12d 6668	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
76	69, 75	eqtr4d 2659	. . . . . . . 8 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) )
77	60	cjred 13966	. . . . . . . 8 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( * ` ( x - C ) ) = ( x - C ) )
78	76, 77	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( ( * ` ( ( F ` x ) - ( F ` C ) ) ) / ( * ` ( x - C ) ) ) = ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) )
79	67, 78	eqtrd 2656	. . . . . 6 |- ( ( ph /\ x e. ( X \ { C } ) ) -> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) )
80	79	mpteq2dva 4744	. . . . 5 |- ( ph -> ( x e. ( X \ { C } ) |-> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) = ( x e. ( X \ { C } ) |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) )
81	48, 80	eqtrd 2656	. . . 4 |- ( ph -> ( * o. ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) = ( x e. ( X \ { C } ) |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) )
82	81	oveq1d 6665	. . 3 |- ( ph -> ( ( * o. ( x e. ( X \ { C } ) |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) lim CC C ) = ( ( x e. ( X \ { C } ) |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) lim CC C ) )
83	42, 82	eleqtrd 2703	. 2 |- ( ph -> ( * ` ( ( RR _D F ) ` C ) ) e. ( ( x e. ( X \ { C } ) |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) lim CC C ) )
84		eqid 2622	. . 3 |- ( x e. ( X \ { C } ) |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) = ( x e. ( X \ { C } ) |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) )
85		fco 6058	. . . 4 |- ( ( * : CC --> CC /\ F : X --> CC ) -> ( * o. F ) : X --> CC )
86	44, 3, 85	sylancr 695	. . 3 |- ( ph -> ( * o. F ) : X --> CC )
87	6, 5, 84, 2, 86, 4	eldv 23662	. 2 |- ( ph -> ( C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) <-> ( C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) /\ ( * ` ( ( RR _D F ) ` C ) ) e. ( ( x e. ( X \ { C } ) |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) lim CC C ) ) ) )
88	9, 83, 87	mpbir2and 957	1 |- ( ph -> C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   - cmin 10266   / cdiv 10684   (,)cioo 12175   *ccj 13836   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746  TopOnctopon 20715   intcnt 20821   Cn ccn 21028   CnP ccnp 21029   -cn->ccncf 22679   lim CC climc 23626   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvcj  23713