Theorem dvcjbr 15382

Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 15383. (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 8091	. . . . 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 2229	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
6	5	tgioo2cntop 15231	. . . 4 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t RR )
7	2, 3, 4, 6, 5	dvbssntrcntop 15358	. . 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 3225	. 2 |- ( ph -> C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) )
10	4, 1	sstrdi 3236	. . . . . 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 15359	. . . . . . . 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 3225	. . . . . 6 |- ( ph -> C e. X )
17	3, 10, 16	dvlemap 15354	. . . . 5 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) e. CC )
18	17	fmpttd 5790	. . . 4 |- ( ph -> ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) : { w e. X | w # C } --> CC )
19		ssidd 3245	. . . 4 |- ( ph -> CC C_ CC )
20	5	cntoptopon 15206	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
21	20	toponrestid 14695	. . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
22	3	fdmd 5480	. . . . . . . . . . . . 13 |- ( ph -> dom F = X )
23	22	feq2d 5461	. . . . . . . . . . . 12 |- ( ph -> ( F : dom F --> CC <-> F : X --> CC ) )
24	3, 23	mpbird 167	. . . . . . . . . . 11 |- ( ph -> F : dom F --> CC )
25	22, 4	eqsstrd 3260	. . . . . . . . . . 11 |- ( ph -> dom F C_ RR )
26		cnex 8123	. . . . . . . . . . . 12 |- CC e. _V
27		reex 8133	. . . . . . . . . . . 12 |- RR e. _V
28	26, 27	elpm2 6827	. . . . . . . . . . 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 15363	. . . . . . . . . 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 5477	. . . . . . . 8 |- ( ph -> Fun ( RR _D F ) )
33		funfvbrb 5748	. . . . . . . 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 2229	. . . . . . 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 15356	. . . . . 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 15262	. . . . . 6 |- * e. ( CC -cn-> CC )
41	5	cncfcn1cntop 15268	. . . . . 6 |- ( CC -cn-> CC ) = ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
42	40, 41	eleqtri 2304	. . . . 5 |- * e. ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
43	31, 8	ffvelcdmd 5771	. . . . 5 |- ( ph -> ( ( RR _D F ) ` C ) e. CC )
44		unicntopcntop 15216	. . . . . 6 |- CC = U. ( MetOpen ` ( abs o. - ) )
45	44	cncnpi 14902	. . . . 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 15349	. . 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 11358	. . . . . . 7 |- * : CC --> CC
49	48	a1i 9	. . . . . 6 |- ( ph -> * : CC --> CC )
50	49, 17	cofmpt 5804	. . . . 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 2956	. . . . . . . . . . 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 5771	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( F ` x ) e. CC )
55	3, 16	ffvelcdmd 5771	. . . . . . . . . 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 8457	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( F ` x ) - ( F ` C ) ) e. CC )
58	4	sselda 3224	. . . . . . . . . . 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 3225	. . . . . . . . . . 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 8527	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. RR )
63	62	recnd 8175	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. CC )
64	59	recnd 8175	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> x e. CC )
65	61	recnd 8175	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> C e. CC )
66		breq1 4086	. . . . . . . . . . . 12 |- ( w = x -> ( w # C <-> x # C ) )
67	66	elrab 2959	. . . . . . . . . . 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 8791	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) # 0 )
71	57, 63, 70	cjdivapd 11479	. . . . . . 7 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( ( * ` ( ( F ` x ) - ( F ` C ) ) ) / ( * ` ( x - C ) ) ) )
72		cjsub 11403	. . . . . . . . . 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 5705	. . . . . . . . . . 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 5705	. . . . . . . . . . . 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 6019	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
80	73, 79	eqtr4d 2265	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) )
81	62	cjred 11482	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( x - C ) ) = ( x - C ) )
82	80, 81	oveq12d 6019	. . . . . . 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 2262	. . . . . 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 4174	. . . . 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 2262	. . . 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 6016	. . 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 2308	. 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 2229	. . 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 5489	. . . 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 15356	. 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 950	1 |- ( ph -> C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   C_ wss 3197   class class class wbr 4083   |-> cmpt 4145   dom cdm 4719   ran crn 4720   o. ccom 4723   Fun wfun 5312   -->wf 5314   ` cfv 5318  (class class class)co 6001   ^pm cpm 6796   CCcc 7997   RRcr 7998   - cmin 8317   # cap 8728   / cdiv 8819   (,)cioo 10084   *ccj 11350   abscabs 11508   topGenctg 13287   MetOpencmopn 14505   intcnt 14767   Cn ccn 14859   CnP ccnp 14860   -cn->ccncf 15244   lim CC climc 15328   _D cdv 15329

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-map 6797  df-pm 6798  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-xneg 9968  df-xadd 9969  df-ioo 10088  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-rest 13274  df-topgen 13293  df-psmet 14507  df-xmet 14508  df-met 14509  df-bl 14510  df-mopn 14511  df-top 14672  df-topon 14685  df-bases 14717  df-ntr 14770  df-cn 14862  df-cnp 14863  df-cncf 15245  df-limced 15330  df-dvap 15331

This theorem is referenced by:  dvcj  15383