Theorem dvcjbr 15699

Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 15700. (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 8235	. . . . 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 2234	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
6	5	tgioo2cntop 15548	. . . 4 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t RR )
7	2, 3, 4, 6, 5	dvbssntrcntop 15675	. . 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 3243	. 2 |- ( ph -> C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) )
10	4, 1	sstrdi 3254	. . . . . 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 15676	. . . . . . . 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 3243	. . . . . 6 |- ( ph -> C e. X )
17	3, 10, 16	dvlemap 15671	. . . . 5 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) e. CC )
18	17	fmpttd 5837	. . . 4 |- ( ph -> ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) : { w e. X | w # C } --> CC )
19		ssidd 3263	. . . 4 |- ( ph -> CC C_ CC )
20	5	cntoptopon 15523	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
21	20	toponrestid 15012	. . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
22	3	fdmd 5520	. . . . . . . . . . . . 13 |- ( ph -> dom F = X )
23	22	feq2d 5501	. . . . . . . . . . . 12 |- ( ph -> ( F : dom F --> CC <-> F : X --> CC ) )
24	3, 23	mpbird 167	. . . . . . . . . . 11 |- ( ph -> F : dom F --> CC )
25	22, 4	eqsstrd 3278	. . . . . . . . . . 11 |- ( ph -> dom F C_ RR )
26		cnex 8267	. . . . . . . . . . . 12 |- CC e. _V
27		reex 8277	. . . . . . . . . . . 12 |- RR e. _V
28	26, 27	elpm2 6927	. . . . . . . . . . 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 15680	. . . . . . . . . 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 5517	. . . . . . . 8 |- ( ph -> Fun ( RR _D F ) )
33		funfvbrb 5796	. . . . . . . 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 2234	. . . . . . 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 15673	. . . . . 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 15579	. . . . . 6 |- * e. ( CC -cn-> CC )
41	5	cncfcn1cntop 15585	. . . . . 6 |- ( CC -cn-> CC ) = ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
42	40, 41	eleqtri 2309	. . . . 5 |- * e. ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
43	31, 8	ffvelcdmd 5818	. . . . 5 |- ( ph -> ( ( RR _D F ) ` C ) e. CC )
44		unicntopcntop 15533	. . . . . 6 |- CC = U. ( MetOpen ` ( abs o. - ) )
45	44	cncnpi 15219	. . . . 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 15666	. . 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 11557	. . . . . . 7 |- * : CC --> CC
49	48	a1i 9	. . . . . 6 |- ( ph -> * : CC --> CC )
50	49, 17	cofmpt 5851	. . . . 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 2973	. . . . . . . . . . 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 5818	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( F ` x ) e. CC )
55	3, 16	ffvelcdmd 5818	. . . . . . . . . 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 8600	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( F ` x ) - ( F ` C ) ) e. CC )
58	4	sselda 3242	. . . . . . . . . . 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 3243	. . . . . . . . . . 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 8671	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. RR )
63	62	recnd 8318	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. CC )
64	59	recnd 8318	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> x e. CC )
65	61	recnd 8318	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> C e. CC )
66		breq1 4117	. . . . . . . . . . . 12 |- ( w = x -> ( w # C <-> x # C ) )
67	66	elrab 2976	. . . . . . . . . . 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 8935	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) # 0 )
71	57, 63, 70	cjdivapd 11678	. . . . . . 7 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( ( * ` ( ( F ` x ) - ( F ` C ) ) ) / ( * ` ( x - C ) ) ) )
72		cjsub 11602	. . . . . . . . . 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 5753	. . . . . . . . . . 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 5753	. . . . . . . . . . . 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 6076	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
80	73, 79	eqtr4d 2270	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) )
81	62	cjred 11681	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( x - C ) ) = ( x - C ) )
82	80, 81	oveq12d 6076	. . . . . . 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 2267	. . . . . 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 4205	. . . . 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 2267	. . . 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 6073	. . 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 2313	. 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 2234	. . 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 5532	. . . 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 15673	. 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 953	1 |- ( ph -> C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {crab 2526   C_ wss 3214   class class class wbr 4114   |-> cmpt 4176   dom cdm 4754   ran crn 4755   o. ccom 4758   Fun wfun 5351   -->wf 5353   ` cfv 5357  (class class class)co 6058   ^pm cpm 6896   CCcc 8141   RRcr 8142   - cmin 8460   # cap 8872   / cdiv 8963   (,)cioo 10240   *ccj 11549   abscabs 11707   topGenctg 13551   MetOpencmopn 14815   intcnt 15084   Cn ccn 15176   CnP ccnp 15177   -cn->ccncf 15561   lim CC climc 15645   _D cdv 15646

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-map 6897  df-pm 6898  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-ioo 10244  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-rest 13538  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820  df-mopn 14821  df-top 14989  df-topon 15002  df-bases 15034  df-ntr 15087  df-cn 15179  df-cnp 15180  df-cncf 15562  df-limced 15647  df-dvap 15648

This theorem is referenced by:  dvcj  15700