Theorem dvcjbr 14468

Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 14469. (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 w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-resscn 7917	. . . . 5 |- RR C_ CC
2	1	a1i 9	. . . 4 |- ( ph -> RR C_ CC )
3		dvcj.f	. . . 4 |- ( ph -> F : X --> CC )
4		dvcj.x	. . . 4 |- ( ph -> X C_ RR )
5		eqid 2187	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
6	5	tgioo2cntop 14345	. . . 4 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t RR )
7	2, 3, 4, 6, 5	dvbssntrcntop 14449	. . 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 3168	. 2 |- ( ph -> C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) )
10	4, 1	sstrdi 3179	. . . . . 6 |- ( ph -> X C_ CC )
11	1	a1i 9	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> RR C_ CC )
12		simpl 109	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> F : X --> CC )
13		simpr 110	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> X C_ RR )
14	11, 12, 13	dvbss 14450	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D F ) C_ X )
15	3, 4, 14	syl2anc 411	. . . . . . 7 |- ( ph -> dom ( RR _D F ) C_ X )
16	15, 8	sseldd 3168	. . . . . 6 |- ( ph -> C e. X )
17	3, 10, 16	dvlemap 14445	. . . . 5 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) e. CC )
18	17	fmpttd 5684	. . . 4 |- ( ph -> ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) : { w e. X | w # C } --> CC )
19		ssidd 3188	. . . 4 |- ( ph -> CC C_ CC )
20	5	cntoptopon 14328	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
21	20	toponrestid 13817	. . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
22	3	fdmd 5384	. . . . . . . . . . . . 13 |- ( ph -> dom F = X )
23	22	feq2d 5365	. . . . . . . . . . . 12 |- ( ph -> ( F : dom F --> CC <-> F : X --> CC ) )
24	3, 23	mpbird 167	. . . . . . . . . . 11 |- ( ph -> F : dom F --> CC )
25	22, 4	eqsstrd 3203	. . . . . . . . . . 11 |- ( ph -> dom F C_ RR )
26		cnex 7949	. . . . . . . . . . . 12 |- CC e. _V
27		reex 7959	. . . . . . . . . . . 12 |- RR e. _V
28	26, 27	elpm2 6694	. . . . . . . . . . 11 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
29	24, 25, 28	sylanbrc 417	. . . . . . . . . 10 |- ( ph -> F e. ( CC ^pm RR ) )
30		dvfpm 14454	. . . . . . . . . 10 |- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
31	29, 30	syl 14	. . . . . . . . 9 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> CC )
32	31	ffund 5381	. . . . . . . 8 |- ( ph -> Fun ( RR _D F ) )
33		funfvbrb 5642	. . . . . . . 8 |- ( Fun ( RR _D F ) -> ( C e. dom ( RR _D F ) <-> C ( RR _D F ) ( ( RR _D F ) ` C ) ) )
34	32, 33	syl 14	. . . . . . 7 |- ( ph -> ( C e. dom ( RR _D F ) <-> C ( RR _D F ) ( ( RR _D F ) ` C ) ) )
35	8, 34	mpbid 147	. . . . . 6 |- ( ph -> C ( RR _D F ) ( ( RR _D F ) ` C ) )
36		eqid 2187	. . . . . . 7 |- ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) )
37	6, 5, 36, 2, 3, 4	eldvap 14447	. . . . . 6 |- ( ph -> ( C ( RR _D F ) ( ( RR _D F ) ` C ) <-> ( C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) /\ ( ( RR _D F ) ` C ) e. ( ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) lim CC C ) ) ) )
38	35, 37	mpbid 147	. . . . 5 |- ( ph -> ( C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) /\ ( ( RR _D F ) ` C ) e. ( ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) lim CC C ) ) )
39	38	simprd 114	. . . 4 |- ( ph -> ( ( RR _D F ) ` C ) e. ( ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) lim CC C ) )
40		cjcncf 14371	. . . . . 6 |- * e. ( CC -cn-> CC )
41	5	cncfcn1cntop 14377	. . . . . 6 |- ( CC -cn-> CC ) = ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
42	40, 41	eleqtri 2262	. . . . 5 |- * e. ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
43	31, 8	ffvelcdmd 5665	. . . . 5 |- ( ph -> ( ( RR _D F ) ` C ) e. CC )
44		unicntopcntop 14332	. . . . . 6 |- CC = U. ( MetOpen ` ( abs o. - ) )
45	44	cncnpi 14024	. . . . 5 |- ( ( * e. ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) ) /\ ( ( RR _D F ) ` C ) e. CC ) -> * e. ( ( ( MetOpen ` ( abs o. - ) ) CnP ( MetOpen ` ( abs o. - ) ) ) ` ( ( RR _D F ) ` C ) ) )
46	42, 43, 45	sylancr 414	. . . 4 |- ( ph -> * e. ( ( ( MetOpen ` ( abs o. - ) ) CnP ( MetOpen ` ( abs o. - ) ) ) ` ( ( RR _D F ) ` C ) ) )
47	18, 19, 5, 21, 39, 46	limccnpcntop 14440	. . 3 |- ( ph -> ( * ` ( ( RR _D F ) ` C ) ) e. ( ( * o. ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) lim CC C ) )
48		cjf 10870	. . . . . . 7 |- * : CC --> CC
49	48	a1i 9	. . . . . 6 |- ( ph -> * : CC --> CC )
50	49, 17	cofmpt 5698	. . . . 5 |- ( ph -> ( * o. ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) = ( x e. { w e. X | w # C } |-> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) )
51	3	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ x e. { w e. X | w # C } ) -> F : X --> CC )
52		elrabi 2902	. . . . . . . . . . 11 |- ( x e. { w e. X | w # C } -> x e. X )
53	52	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ x e. { w e. X | w # C } ) -> x e. X )
54	51, 53	ffvelcdmd 5665	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( F ` x ) e. CC )
55	3, 16	ffvelcdmd 5665	. . . . . . . . . 10 |- ( ph -> ( F ` C ) e. CC )
56	55	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( F ` C ) e. CC )
57	54, 56	subcld 8282	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( F ` x ) - ( F ` C ) ) e. CC )
58	4	sselda 3167	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> x e. RR )
59	52, 58	sylan2 286	. . . . . . . . . 10 |- ( ( ph /\ x e. { w e. X | w # C } ) -> x e. RR )
60	4, 16	sseldd 3168	. . . . . . . . . . 11 |- ( ph -> C e. RR )
61	60	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ x e. { w e. X | w # C } ) -> C e. RR )
62	59, 61	resubcld 8352	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. RR )
63	62	recnd 8000	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. CC )
64	59	recnd 8000	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> x e. CC )
65	61	recnd 8000	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> C e. CC )
66		breq1 4018	. . . . . . . . . . . 12 |- ( w = x -> ( w # C <-> x # C ) )
67	66	elrab 2905	. . . . . . . . . . 11 |- ( x e. { w e. X | w # C } <-> ( x e. X /\ x # C ) )
68	67	simprbi 275	. . . . . . . . . 10 |- ( x e. { w e. X | w # C } -> x # C )
69	68	adantl 277	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> x # C )
70	64, 65, 69	subap0d 8615	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) # 0 )
71	57, 63, 70	cjdivapd 10991	. . . . . . 7 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( ( * ` ( ( F ` x ) - ( F ` C ) ) ) / ( * ` ( x - C ) ) ) )
72		cjsub 10915	. . . . . . . . . 10 |- ( ( ( F ` x ) e. CC /\ ( F ` C ) e. CC ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
73	54, 56, 72	syl2anc 411	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
74		fvco3 5600	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ x e. X ) -> ( ( * o. F ) ` x ) = ( * ` ( F ` x ) ) )
75	3, 52, 74	syl2an 289	. . . . . . . . . 10 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( * o. F ) ` x ) = ( * ` ( F ` x ) ) )
76		fvco3 5600	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ C e. X ) -> ( ( * o. F ) ` C ) = ( * ` ( F ` C ) ) )
77	3, 16, 76	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( ( * o. F ) ` C ) = ( * ` ( F ` C ) ) )
78	77	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( * o. F ) ` C ) = ( * ` ( F ` C ) ) )
79	75, 78	oveq12d 5906	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
80	73, 79	eqtr4d 2223	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) )
81	62	cjred 10994	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( x - C ) ) = ( x - C ) )
82	80, 81	oveq12d 5906	. . . . . . 7 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( * ` ( ( F ` x ) - ( F ` C ) ) ) / ( * ` ( x - C ) ) ) = ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) )
83	71, 82	eqtrd 2220	. . . . . 6 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) )
84	83	mpteq2dva 4105	. . . . 5 |- ( ph -> ( x e. { w e. X | w # C } |-> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) = ( x e. { w e. X | w # C } |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) )
85	50, 84	eqtrd 2220	. . . 4 |- ( ph -> ( * o. ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) = ( x e. { w e. X | w # C } |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) )
86	85	oveq1d 5903	. . 3 |- ( ph -> ( ( * o. ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) ) lim CC C ) = ( ( x e. { w e. X | w # C } |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) lim CC C ) )
87	47, 86	eleqtrd 2266	. 2 |- ( ph -> ( * ` ( ( RR _D F ) ` C ) ) e. ( ( x e. { w e. X | w # C } |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) lim CC C ) )
88		eqid 2187	. . 3 |- ( x e. { w e. X | w # C } |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) = ( x e. { w e. X | w # C } |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) )
89		fco 5393	. . . 4 |- ( ( * : CC --> CC /\ F : X --> CC ) -> ( * o. F ) : X --> CC )
90	48, 3, 89	sylancr 414	. . 3 |- ( ph -> ( * o. F ) : X --> CC )
91	6, 5, 88, 2, 90, 4	eldvap 14447	. 2 |- ( ph -> ( C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) <-> ( C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) /\ ( * ` ( ( RR _D F ) ` C ) ) e. ( ( x e. { w e. X | w # C } |-> ( ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) / ( x - C ) ) ) lim CC C ) ) ) )
92	9, 87, 91	mpbir2and 945	1 |- ( ph -> C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   {crab 2469   C_ wss 3141   class class class wbr 4015   |-> cmpt 4076   dom cdm 4638   ran crn 4639   o. ccom 4642   Fun wfun 5222   -->wf 5224   ` cfv 5228  (class class class)co 5888   ^pm cpm 6663   CCcc 7823   RRcr 7824   - cmin 8142   # cap 8552   / cdiv 8643   (,)cioo 9902   *ccj 10862   abscabs 11020   topGenctg 12721   MetOpencmopn 13727   intcnt 13889   Cn ccn 13981   CnP ccnp 13982   -cn->ccncf 14353   lim CC climc 14419   _D cdv 14420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-map 6664  df-pm 6665  df-sup 6997  df-inf 6998  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-xneg 9786  df-xadd 9787  df-ioo 9906  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-rest 12708  df-topgen 12727  df-psmet 13729  df-xmet 13730  df-met 13731  df-bl 13732  df-mopn 13733  df-top 13794  df-topon 13807  df-bases 13839  df-ntr 13892  df-cn 13984  df-cnp 13985  df-cncf 14354  df-limced 14421  df-dvap 14422

This theorem is referenced by:  dvcj  14469