Theorem dvcjbr 15431

Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 15432. (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 8123	. . . . 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 2231	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
6	5	tgioo2cntop 15280	. . . 4 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t RR )
7	2, 3, 4, 6, 5	dvbssntrcntop 15407	. . 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 3228	. 2 |- ( ph -> C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) )
10	4, 1	sstrdi 3239	. . . . . 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 15408	. . . . . . . 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 3228	. . . . . 6 |- ( ph -> C e. X )
17	3, 10, 16	dvlemap 15403	. . . . 5 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) e. CC )
18	17	fmpttd 5802	. . . 4 |- ( ph -> ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) : { w e. X | w # C } --> CC )
19		ssidd 3248	. . . 4 |- ( ph -> CC C_ CC )
20	5	cntoptopon 15255	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
21	20	toponrestid 14744	. . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
22	3	fdmd 5489	. . . . . . . . . . . . 13 |- ( ph -> dom F = X )
23	22	feq2d 5470	. . . . . . . . . . . 12 |- ( ph -> ( F : dom F --> CC <-> F : X --> CC ) )
24	3, 23	mpbird 167	. . . . . . . . . . 11 |- ( ph -> F : dom F --> CC )
25	22, 4	eqsstrd 3263	. . . . . . . . . . 11 |- ( ph -> dom F C_ RR )
26		cnex 8155	. . . . . . . . . . . 12 |- CC e. _V
27		reex 8165	. . . . . . . . . . . 12 |- RR e. _V
28	26, 27	elpm2 6848	. . . . . . . . . . 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 15412	. . . . . . . . . 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 5486	. . . . . . . 8 |- ( ph -> Fun ( RR _D F ) )
33		funfvbrb 5760	. . . . . . . 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 2231	. . . . . . 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 15405	. . . . . 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 15311	. . . . . 6 |- * e. ( CC -cn-> CC )
41	5	cncfcn1cntop 15317	. . . . . 6 |- ( CC -cn-> CC ) = ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
42	40, 41	eleqtri 2306	. . . . 5 |- * e. ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
43	31, 8	ffvelcdmd 5783	. . . . 5 |- ( ph -> ( ( RR _D F ) ` C ) e. CC )
44		unicntopcntop 15265	. . . . . 6 |- CC = U. ( MetOpen ` ( abs o. - ) )
45	44	cncnpi 14951	. . . . 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 15398	. . 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 11407	. . . . . . 7 |- * : CC --> CC
49	48	a1i 9	. . . . . 6 |- ( ph -> * : CC --> CC )
50	49, 17	cofmpt 5816	. . . . 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 2959	. . . . . . . . . . 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 5783	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( F ` x ) e. CC )
55	3, 16	ffvelcdmd 5783	. . . . . . . . . 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 8489	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( F ` x ) - ( F ` C ) ) e. CC )
58	4	sselda 3227	. . . . . . . . . . 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 3228	. . . . . . . . . . 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 8559	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. RR )
63	62	recnd 8207	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. CC )
64	59	recnd 8207	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> x e. CC )
65	61	recnd 8207	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> C e. CC )
66		breq1 4091	. . . . . . . . . . . 12 |- ( w = x -> ( w # C <-> x # C ) )
67	66	elrab 2962	. . . . . . . . . . 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 8823	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) # 0 )
71	57, 63, 70	cjdivapd 11528	. . . . . . 7 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( ( * ` ( ( F ` x ) - ( F ` C ) ) ) / ( * ` ( x - C ) ) ) )
72		cjsub 11452	. . . . . . . . . 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 5717	. . . . . . . . . . 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 5717	. . . . . . . . . . . 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 6035	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
80	73, 79	eqtr4d 2267	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) )
81	62	cjred 11531	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( x - C ) ) = ( x - C ) )
82	80, 81	oveq12d 6035	. . . . . . 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 2264	. . . . . 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 4179	. . . . 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 2264	. . . 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 6032	. . 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 2310	. 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 2231	. . 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 5500	. . . 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 15405	. 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 952	1 |- ( ph -> C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {crab 2514   C_ wss 3200   class class class wbr 4088   |-> cmpt 4150   dom cdm 4725   ran crn 4726   o. ccom 4729   Fun wfun 5320   -->wf 5322   ` cfv 5326  (class class class)co 6017   ^pm cpm 6817   CCcc 8029   RRcr 8030   - cmin 8349   # cap 8760   / cdiv 8851   (,)cioo 10122   *ccj 11399   abscabs 11557   topGenctg 13336   MetOpencmopn 14554   intcnt 14816   Cn ccn 14908   CnP ccnp 14909   -cn->ccncf 15293   lim CC climc 15377   _D cdv 15378

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-map 6818  df-pm 6819  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-xneg 10006  df-xadd 10007  df-ioo 10126  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-rest 13323  df-topgen 13342  df-psmet 14556  df-xmet 14557  df-met 14558  df-bl 14559  df-mopn 14560  df-top 14721  df-topon 14734  df-bases 14766  df-ntr 14819  df-cn 14911  df-cnp 14912  df-cncf 15294  df-limced 15379  df-dvap 15380

This theorem is referenced by:  dvcj  15432