Theorem dvcjbr 15934

Description: The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 15935. (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 8277	. . . . 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 2238	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
6	5	tgioo2cntop 15783	. . . 4 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t RR )
7	2, 3, 4, 6, 5	dvbssntrcntop 15910	. . 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 3249	. 2 |- ( ph -> C e. ( ( int ` ( topGen ` ran (,) ) ) ` X ) )
10	4, 1	sstrdi 3260	. . . . . 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 15911	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D F ) C_ X )
15	3, 4, 14	syl2anc 415	. . . . . . 7 |- ( ph -> dom ( RR _D F ) C_ X )
16	15, 8	sseldd 3249	. . . . . 6 |- ( ph -> C e. X )
17	3, 10, 16	dvlemap 15906	. . . . 5 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) e. CC )
18	17	fmpttd 5866	. . . 4 |- ( ph -> ( x e. { w e. X | w # C } |-> ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) : { w e. X | w # C } --> CC )
19		ssidd 3269	. . . 4 |- ( ph -> CC C_ CC )
20	5	cntoptopon 15758	. . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
21	20	toponrestid 15247	. . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
22	3	fdmd 5543	. . . . . . . . . . . . 13 |- ( ph -> dom F = X )
23	22	feq2d 5524	. . . . . . . . . . . 12 |- ( ph -> ( F : dom F --> CC <-> F : X --> CC ) )
24	3, 23	mpbird 167	. . . . . . . . . . 11 |- ( ph -> F : dom F --> CC )
25	22, 4	eqsstrd 3284	. . . . . . . . . . 11 |- ( ph -> dom F C_ RR )
26		cnex 8309	. . . . . . . . . . . 12 |- CC e. _V
27		reex 8319	. . . . . . . . . . . 12 |- RR e. _V
28	26, 27	elpm2 6965	. . . . . . . . . . 11 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
29	24, 25, 28	sylanbrc 421	. . . . . . . . . 10 |- ( ph -> F e. ( CC ^pm RR ) )
30		dvfpm 15915	. . . . . . . . . 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 5540	. . . . . . . 8 |- ( ph -> Fun ( RR _D F ) )
33		funfvbrb 5825	. . . . . . . 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 2238	. . . . . . 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 15908	. . . . . 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 15814	. . . . . 6 |- * e. ( CC -cn-> CC )
41	5	cncfcn1cntop 15820	. . . . . 6 |- ( CC -cn-> CC ) = ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
42	40, 41	eleqtri 2313	. . . . 5 |- * e. ( ( MetOpen ` ( abs o. - ) ) Cn ( MetOpen ` ( abs o. - ) ) )
43	31, 8	ffvelcdmd 5847	. . . . 5 |- ( ph -> ( ( RR _D F ) ` C ) e. CC )
44		unicntopcntop 15768	. . . . . 6 |- CC = U. ( MetOpen ` ( abs o. - ) )
45	44	cncnpi 15454	. . . . 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 418	. . . 4 |- ( ph -> * e. ( ( ( MetOpen ` ( abs o. - ) ) CnP ( MetOpen ` ( abs o. - ) ) ) ` ( ( RR _D F ) ` C ) ) )
47	18, 19, 5, 21, 39, 46	limccnpcntop 15901	. . 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 11634	. . . . . . 7 |- * : CC --> CC
49	48	a1i 9	. . . . . 6 |- ( ph -> * : CC --> CC )
50	49, 17	cofmpt 5880	. . . . 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 2979	. . . . . . . . . . 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 5847	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( F ` x ) e. CC )
55	3, 16	ffvelcdmd 5847	. . . . . . . . . 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 8644	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( F ` x ) - ( F ` C ) ) e. CC )
58	4	sselda 3248	. . . . . . . . . . 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 3249	. . . . . . . . . . 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 8715	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. RR )
63	62	recnd 8360	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) e. CC )
64	59	recnd 8360	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> x e. CC )
65	61	recnd 8360	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> C e. CC )
66		breq1 4134	. . . . . . . . . . . 12 |- ( w = x -> ( w # C <-> x # C ) )
67	66	elrab 2982	. . . . . . . . . . 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 8980	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( x - C ) # 0 )
71	57, 63, 70	cjdivapd 11756	. . . . . . 7 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( ( F ` x ) - ( F ` C ) ) / ( x - C ) ) ) = ( ( * ` ( ( F ` x ) - ( F ` C ) ) ) / ( * ` ( x - C ) ) ) )
72		cjsub 11679	. . . . . . . . . 10 |- ( ( ( F ` x ) e. CC /\ ( F ` C ) e. CC ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
73	54, 56, 72	syl2anc 415	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
74		fvco3 5779	. . . . . . . . . . 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 5779	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ C e. X ) -> ( ( * o. F ) ` C ) = ( * ` ( F ` C ) ) )
77	3, 16, 76	syl2anc 415	. . . . . . . . . . 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 6106	. . . . . . . . 9 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) = ( ( * ` ( F ` x ) ) - ( * ` ( F ` C ) ) ) )
80	73, 79	eqtr4d 2274	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( ( F ` x ) - ( F ` C ) ) ) = ( ( ( * o. F ) ` x ) - ( ( * o. F ) ` C ) ) )
81	62	cjred 11759	. . . . . . . 8 |- ( ( ph /\ x e. { w e. X | w # C } ) -> ( * ` ( x - C ) ) = ( x - C ) )
82	80, 81	oveq12d 6106	. . . . . . 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 2271	. . . . . 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 4223	. . . . 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 2271	. . . 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 6103	. . 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 2317	. 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 2238	. . 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 5555	. . . 4 |- ( ( * : CC --> CC /\ F : X --> CC ) -> ( * o. F ) : X --> CC )
90	48, 3, 89	sylancr 418	. . 3 |- ( ph -> ( * o. F ) : X --> CC )
91	6, 5, 88, 2, 90, 4	eldvap 15908	. 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 957	1 |- ( ph -> C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220   class class class wbr 4131   |-> cmpt 4193   dom cdm 4776   ran crn 4777   o. ccom 4780   Fun wfun 5374   -->wf 5376   ` cfv 5380  (class class class)co 6088   ^pm cpm 6927   CCcc 8183   RRcr 8184   - cmin 8504   # cap 8917   / cdiv 9010   (,)cioo 10307   *ccj 11626   abscabs 11785   topGenctg 13667   MetOpencmopn 14987   intcnt 15319   Cn ccn 15411   CnP ccnp 15412   -cn->ccncf 15796   lim CC climc 15880   _D cdv 15881

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-map 6928  df-pm 6929  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-xneg 10191  df-xadd 10192  df-ioo 10311  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-rest 13654  df-topgen 13673  df-psmet 14989  df-xmet 14990  df-met 14991  df-bl 14992  df-mopn 14993  df-top 15224  df-topon 15237  df-bases 15269  df-ntr 15322  df-cn 15414  df-cnp 15415  df-cncf 15797  df-limced 15882  df-dvap 15883

This theorem is used by:  dvcj  15935