Theorem dvcj 23713

Description: The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 23712. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Assertion

Ref	Expression
dvcj	|- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )

Proof of Theorem dvcj

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvf 23671	. . . . 5 |- ( RR _D ( * o. F ) ) : dom ( RR _D ( * o. F ) ) --> CC
2		ffun 6048	. . . . 5 |- ( ( RR _D ( * o. F ) ) : dom ( RR _D ( * o. F ) ) --> CC -> Fun ( RR _D ( * o. F ) ) )
3	1, 2	ax-mp 5	. . . 4 |- Fun ( RR _D ( * o. F ) )
4		simpll 790	. . . . 5 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> F : X --> CC )
5		simplr 792	. . . . 5 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> X C_ RR )
6		simpr 477	. . . . 5 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x e. dom ( RR _D F ) )
7	4, 5, 6	dvcjbr 23712	. . . 4 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) )
8		funbrfv 6234	. . . 4 |- ( Fun ( RR _D ( * o. F ) ) -> ( x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) -> ( ( RR _D ( * o. F ) ) ` x ) = ( * ` ( ( RR _D F ) ` x ) ) ) )
9	3, 7, 8	mpsyl 68	. . 3 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D ( * o. F ) ) ` x ) = ( * ` ( ( RR _D F ) ` x ) ) )
10	9	mpteq2dva 4744	. 2 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. dom ( RR _D F ) |-> ( ( RR _D ( * o. F ) ) ` x ) ) = ( x e. dom ( RR _D F ) |-> ( * ` ( ( RR _D F ) ` x ) ) ) )
11		cjf 13844	. . . . . . . . . . . . 13 |- * : CC --> CC
12		fco 6058	. . . . . . . . . . . . 13 |- ( ( * : CC --> CC /\ F : X --> CC ) -> ( * o. F ) : X --> CC )
13	11, 12	mpan 706	. . . . . . . . . . . 12 |- ( F : X --> CC -> ( * o. F ) : X --> CC )
14	13	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( * o. F ) : X --> CC )
15		simplr 792	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> X C_ RR )
16		simpr 477	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> x e. dom ( RR _D ( * o. F ) ) )
17	14, 15, 16	dvcjbr 23712	. . . . . . . . . 10 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> x ( RR _D ( * o. ( * o. F ) ) ) ( * ` ( ( RR _D ( * o. F ) ) ` x ) ) )
18		vex 3203	. . . . . . . . . . 11 |- x e. _V
19		fvex 6201	. . . . . . . . . . 11 |- ( * ` ( ( RR _D ( * o. F ) ) ` x ) ) e. _V
20	18, 19	breldm 5329	. . . . . . . . . 10 |- ( x ( RR _D ( * o. ( * o. F ) ) ) ( * ` ( ( RR _D ( * o. F ) ) ` x ) ) -> x e. dom ( RR _D ( * o. ( * o. F ) ) ) )
21	17, 20	syl 17	. . . . . . . . 9 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> x e. dom ( RR _D ( * o. ( * o. F ) ) ) )
22	21	ex 450	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. dom ( RR _D ( * o. F ) ) -> x e. dom ( RR _D ( * o. ( * o. F ) ) ) ) )
23	22	ssrdv 3609	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D ( * o. ( * o. F ) ) ) )
24		ffvelrn 6357	. . . . . . . . . . . . 13 |- ( ( F : X --> CC /\ x e. X ) -> ( F ` x ) e. CC )
25	24	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( F ` x ) e. CC )
26	25	cjcjd 13939	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( * ` ( * ` ( F ` x ) ) ) = ( F ` x ) )
27	26	mpteq2dva 4744	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) = ( x e. X |-> ( F ` x ) ) )
28	25	cjcld 13936	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( * ` ( F ` x ) ) e. CC )
29		simpl 473	. . . . . . . . . . . . 13 |- ( ( F : X --> CC /\ X C_ RR ) -> F : X --> CC )
30	29	feqmptd 6249	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> F = ( x e. X |-> ( F ` x ) ) )
31	11	a1i 11	. . . . . . . . . . . . 13 |- ( ( F : X --> CC /\ X C_ RR ) -> * : CC --> CC )
32	31	feqmptd 6249	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> * = ( y e. CC |-> ( * ` y ) ) )
33		fveq2 6191	. . . . . . . . . . . 12 |- ( y = ( F ` x ) -> ( * ` y ) = ( * ` ( F ` x ) ) )
34	25, 30, 32, 33	fmptco 6396	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) = ( x e. X |-> ( * ` ( F ` x ) ) ) )
35		fveq2 6191	. . . . . . . . . . 11 |- ( y = ( * ` ( F ` x ) ) -> ( * ` y ) = ( * ` ( * ` ( F ` x ) ) ) )
36	28, 34, 32, 35	fmptco 6396	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) )
37	27, 36, 30	3eqtr4d 2666	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = F )
38	37	oveq2d 6666	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. ( * o. F ) ) ) = ( RR _D F ) )
39	38	dmeqd 5326	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. ( * o. F ) ) ) = dom ( RR _D F ) )
40	23, 39	sseqtrd 3641	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D F ) )
41		fvex 6201	. . . . . . . . . 10 |- ( * ` ( ( RR _D F ) ` x ) ) e. _V
42	18, 41	breldm 5329	. . . . . . . . 9 |- ( x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) -> x e. dom ( RR _D ( * o. F ) ) )
43	7, 42	syl 17	. . . . . . . 8 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x e. dom ( RR _D ( * o. F ) ) )
44	43	ex 450	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. dom ( RR _D F ) -> x e. dom ( RR _D ( * o. F ) ) ) )
45	44	ssrdv 3609	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D F ) C_ dom ( RR _D ( * o. F ) ) )
46	40, 45	eqssd 3620	. . . . 5 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) = dom ( RR _D F ) )
47	46	feq2d 6031	. . . 4 |- ( ( F : X --> CC /\ X C_ RR ) -> ( ( RR _D ( * o. F ) ) : dom ( RR _D ( * o. F ) ) --> CC <-> ( RR _D ( * o. F ) ) : dom ( RR _D F ) --> CC ) )
48	1, 47	mpbii 223	. . 3 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) : dom ( RR _D F ) --> CC )
49	48	feqmptd 6249	. 2 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) = ( x e. dom ( RR _D F ) |-> ( ( RR _D ( * o. F ) ) ` x ) ) )
50		dvf 23671	. . . . 5 |- ( RR _D F ) : dom ( RR _D F ) --> CC
51	50	ffvelrni 6358	. . . 4 |- ( x e. dom ( RR _D F ) -> ( ( RR _D F ) ` x ) e. CC )
52	51	adantl 482	. . 3 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
53	50	a1i 11	. . . 4 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
54	53	feqmptd 6249	. . 3 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D F ) = ( x e. dom ( RR _D F ) |-> ( ( RR _D F ) ` x ) ) )
55		fveq2 6191	. . 3 |- ( y = ( ( RR _D F ) ` x ) -> ( * ` y ) = ( * ` ( ( RR _D F ) ` x ) ) )
56	52, 54, 32, 55	fmptco 6396	. 2 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( RR _D F ) ) = ( x e. dom ( RR _D F ) |-> ( * ` ( ( RR _D F ) ` x ) ) ) )
57	10, 49, 56	3eqtr4d 2666	1 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   *ccj 13836   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvfre  23714  dvmptcj  23731