Theorem dvcj 14526

Description: The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 14525. (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		simpll 527	. . . . 5 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> F : X --> CC )
2		simplr 528	. . . . 5 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> X C_ RR )
3		simpr 110	. . . . 5 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x e. dom ( RR _D F ) )
4	1, 2, 3	dvcjbr 14525	. . . 4 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) )
5		cjf 10870	. . . . . . . . . . . 12 |- * : CC --> CC
6		fco 5393	. . . . . . . . . . . 12 |- ( ( * : CC --> CC /\ F : X --> CC ) -> ( * o. F ) : X --> CC )
7	5, 6	mpan 424	. . . . . . . . . . 11 |- ( F : X --> CC -> ( * o. F ) : X --> CC )
8	7	adantr 276	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) : X --> CC )
9	7	fdmd 5384	. . . . . . . . . . . 12 |- ( F : X --> CC -> dom ( * o. F ) = X )
10	9	adantr 276	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( * o. F ) = X )
11	10	feq2d 5365	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( ( * o. F ) : dom ( * o. F ) --> CC <-> ( * o. F ) : X --> CC ) )
12	8, 11	mpbird 167	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) : dom ( * o. F ) --> CC )
13		simpr 110	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> X C_ RR )
14	10, 13	eqsstrd 3203	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( * o. F ) C_ RR )
15		cnex 7949	. . . . . . . . . 10 |- CC e. _V
16		reex 7959	. . . . . . . . . 10 |- RR e. _V
17	15, 16	elpm2 6694	. . . . . . . . 9 |- ( ( * o. F ) e. ( CC ^pm RR ) <-> ( ( * o. F ) : dom ( * o. F ) --> CC /\ dom ( * o. F ) C_ RR ) )
18	12, 14, 17	sylanbrc 417	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) e. ( CC ^pm RR ) )
19		dvfpm 14511	. . . . . . . 8 |- ( ( * o. F ) e. ( CC ^pm RR ) -> ( RR _D ( * o. F ) ) : dom ( RR _D ( * o. F ) ) --> CC )
20	18, 19	syl 14	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) : dom ( RR _D ( * o. F ) ) --> CC )
21	20	ffund 5381	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> Fun ( RR _D ( * o. F ) ) )
22		funbrfv 5567	. . . . . 6 |- ( Fun ( RR _D ( * o. F ) ) -> ( x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) -> ( ( RR _D ( * o. F ) ) ` x ) = ( * ` ( ( RR _D F ) ` x ) ) ) )
23	21, 22	syl 14	. . . . 5 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) -> ( ( RR _D ( * o. F ) ) ` x ) = ( * ` ( ( RR _D F ) ` x ) ) ) )
24	23	adantr 276	. . . 4 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) -> ( ( RR _D ( * o. F ) ) ` x ) = ( * ` ( ( RR _D F ) ` x ) ) ) )
25	4, 24	mpd 13	. . 3 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D ( * o. F ) ) ` x ) = ( * ` ( ( RR _D F ) ` x ) ) )
26	25	mpteq2dva 4105	. 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 ) ) ) )
27		vex 2752	. . . . . . . . . 10 |- x e. _V
28	20	ffvelcdmda 5664	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( ( RR _D ( * o. F ) ) ` x ) e. CC )
29	28	cjcld 10963	. . . . . . . . . 10 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( * ` ( ( RR _D ( * o. F ) ) ` x ) ) e. CC )
30	7	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( * o. F ) : X --> CC )
31		simplr 528	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> X C_ RR )
32		simpr 110	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> x e. dom ( RR _D ( * o. F ) ) )
33	30, 31, 32	dvcjbr 14525	. . . . . . . . . 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 ) ) )
34		breldmg 4845	. . . . . . . . . 10 |- ( ( x e. _V /\ ( * ` ( ( RR _D ( * o. F ) ) ` x ) ) e. CC /\ x ( RR _D ( * o. ( * o. F ) ) ) ( * ` ( ( RR _D ( * o. F ) ) ` x ) ) ) -> x e. dom ( RR _D ( * o. ( * o. F ) ) ) )
35	27, 29, 33, 34	mp3an2i 1352	. . . . . . . . 9 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> x e. dom ( RR _D ( * o. ( * o. F ) ) ) )
36	35	ex 115	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. dom ( RR _D ( * o. F ) ) -> x e. dom ( RR _D ( * o. ( * o. F ) ) ) ) )
37	36	ssrdv 3173	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D ( * o. ( * o. F ) ) ) )
38		ffvelcdm 5662	. . . . . . . . . . . . 13 |- ( ( F : X --> CC /\ x e. X ) -> ( F ` x ) e. CC )
39	38	adantlr 477	. . . . . . . . . . . 12 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( F ` x ) e. CC )
40	39	cjcjd 10966	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( * ` ( * ` ( F ` x ) ) ) = ( F ` x ) )
41	40	mpteq2dva 4105	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) = ( x e. X |-> ( F ` x ) ) )
42	39	cjcld 10963	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( * ` ( F ` x ) ) e. CC )
43		simpl 109	. . . . . . . . . . . . 13 |- ( ( F : X --> CC /\ X C_ RR ) -> F : X --> CC )
44	43	feqmptd 5582	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> F = ( x e. X |-> ( F ` x ) ) )
45	5	a1i 9	. . . . . . . . . . . . 13 |- ( ( F : X --> CC /\ X C_ RR ) -> * : CC --> CC )
46	45	feqmptd 5582	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> * = ( y e. CC |-> ( * ` y ) ) )
47		fveq2 5527	. . . . . . . . . . . 12 |- ( y = ( F ` x ) -> ( * ` y ) = ( * ` ( F ` x ) ) )
48	39, 44, 46, 47	fmptco 5695	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) = ( x e. X |-> ( * ` ( F ` x ) ) ) )
49		fveq2 5527	. . . . . . . . . . 11 |- ( y = ( * ` ( F ` x ) ) -> ( * ` y ) = ( * ` ( * ` ( F ` x ) ) ) )
50	42, 48, 46, 49	fmptco 5695	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) )
51	41, 50, 44	3eqtr4d 2230	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = F )
52	51	oveq2d 5904	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. ( * o. F ) ) ) = ( RR _D F ) )
53	52	dmeqd 4841	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. ( * o. F ) ) ) = dom ( RR _D F ) )
54	37, 53	sseqtrd 3205	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D F ) )
55		ffdm 5398	. . . . . . . . . . . . 13 |- ( F : X --> CC -> ( F : dom F --> CC /\ dom F C_ X ) )
56	55	simpld 112	. . . . . . . . . . . 12 |- ( F : X --> CC -> F : dom F --> CC )
57	56	adantr 276	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> F : dom F --> CC )
58		fdm 5383	. . . . . . . . . . . . 13 |- ( F : X --> CC -> dom F = X )
59	58	adantr 276	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> dom F = X )
60	59, 13	eqsstrd 3203	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> dom F C_ RR )
61	15, 16	elpm2 6694	. . . . . . . . . . 11 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
62	57, 60, 61	sylanbrc 417	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> F e. ( CC ^pm RR ) )
63		dvfpm 14511	. . . . . . . . . 10 |- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
64	62, 63	syl 14	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
65	64	ffvelcdmda 5664	. . . . . . . 8 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
66	65	cjcld 10963	. . . . . . 7 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( * ` ( ( RR _D F ) ` x ) ) e. CC )
67		breldmg 4845	. . . . . . 7 |- ( ( x e. _V /\ ( * ` ( ( RR _D F ) ` x ) ) e. CC /\ x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) ) -> x e. dom ( RR _D ( * o. F ) ) )
68	27, 66, 4, 67	mp3an2i 1352	. . . . . 6 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x e. dom ( RR _D ( * o. F ) ) )
69	54, 68	eqelssd 3186	. . . . 5 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) = dom ( RR _D F ) )
70	69	feq2d 5365	. . . 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 ) )
71	20, 70	mpbid 147	. . 3 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) : dom ( RR _D F ) --> CC )
72	71	feqmptd 5582	. 2 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) = ( x e. dom ( RR _D F ) |-> ( ( RR _D ( * o. F ) ) ` x ) ) )
73	64	feqmptd 5582	. . 3 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D F ) = ( x e. dom ( RR _D F ) |-> ( ( RR _D F ) ` x ) ) )
74		fveq2 5527	. . 3 |- ( y = ( ( RR _D F ) ` x ) -> ( * ` y ) = ( * ` ( ( RR _D F ) ` x ) ) )
75	65, 73, 46, 74	fmptco 5695	. 2 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( RR _D F ) ) = ( x e. dom ( RR _D F ) |-> ( * ` ( ( RR _D F ) ` x ) ) ) )
76	26, 72, 75	3eqtr4d 2230	1 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   _Vcvv 2749   C_ wss 3141   class class class wbr 4015   |-> cmpt 4076   dom cdm 4638   o. ccom 4642   Fun wfun 5222   -->wf 5224   ` cfv 5228  (class class class)co 5888   ^pm cpm 6663   CCcc 7823   RRcr 7824   *ccj 10862   _D cdv 14477

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-map 6664  df-pm 6665  df-sup 6997  df-inf 6998  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-xneg 9786  df-xadd 9787  df-ioo 9906  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-rest 12708  df-topgen 12727  df-psmet 13786  df-xmet 13787  df-met 13788  df-bl 13789  df-mopn 13790  df-top 13851  df-topon 13864  df-bases 13896  df-ntr 13949  df-cn 14041  df-cnp 14042  df-cncf 14411  df-limced 14478  df-dvap 14479

This theorem is referenced by:  dvfre  14527  dvmptcjx  14539