Theorem dvcj 12842

Description: The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 12841. (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 518	. . . . 5 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> F : X --> CC )
2		simplr 519	. . . . 5 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> X C_ RR )
3		simpr 109	. . . . 5 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x e. dom ( RR _D F ) )
4	1, 2, 3	dvcjbr 12841	. . . 4 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) )
5		cjf 10619	. . . . . . . . . . . 12 |- * : CC --> CC
6		fco 5288	. . . . . . . . . . . 12 |- ( ( * : CC --> CC /\ F : X --> CC ) -> ( * o. F ) : X --> CC )
7	5, 6	mpan 420	. . . . . . . . . . 11 |- ( F : X --> CC -> ( * o. F ) : X --> CC )
8	7	adantr 274	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) : X --> CC )
9	7	fdmd 5279	. . . . . . . . . . . 12 |- ( F : X --> CC -> dom ( * o. F ) = X )
10	9	adantr 274	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( * o. F ) = X )
11	10	feq2d 5260	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( ( * o. F ) : dom ( * o. F ) --> CC <-> ( * o. F ) : X --> CC ) )
12	8, 11	mpbird 166	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) : dom ( * o. F ) --> CC )
13		simpr 109	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> X C_ RR )
14	10, 13	eqsstrd 3133	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( * o. F ) C_ RR )
15		cnex 7744	. . . . . . . . . 10 |- CC e. _V
16		reex 7754	. . . . . . . . . 10 |- RR e. _V
17	15, 16	elpm2 6574	. . . . . . . . 9 |- ( ( * o. F ) e. ( CC ^pm RR ) <-> ( ( * o. F ) : dom ( * o. F ) --> CC /\ dom ( * o. F ) C_ RR ) )
18	12, 14, 17	sylanbrc 413	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) e. ( CC ^pm RR ) )
19		dvfpm 12827	. . . . . . . 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 5276	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> Fun ( RR _D ( * o. F ) ) )
22		funbrfv 5460	. . . . . 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 274	. . . 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 4018	. 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 2689	. . . . . . . . . 10 |- x e. _V
28	20	ffvelrnda 5555	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( ( RR _D ( * o. F ) ) ` x ) e. CC )
29	28	cjcld 10712	. . . . . . . . . 10 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( * ` ( ( RR _D ( * o. F ) ) ` x ) ) e. CC )
30	7	ad2antrr 479	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( * o. F ) : X --> CC )
31		simplr 519	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> X C_ RR )
32		simpr 109	. . . . . . . . . . 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 12841	. . . . . . . . . 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 4745	. . . . . . . . . 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 1320	. . . . . . . . 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 114	. . . . . . . 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 3103	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D ( * o. ( * o. F ) ) ) )
38		ffvelrn 5553	. . . . . . . . . . . . 13 |- ( ( F : X --> CC /\ x e. X ) -> ( F ` x ) e. CC )
39	38	adantlr 468	. . . . . . . . . . . 12 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( F ` x ) e. CC )
40	39	cjcjd 10715	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( * ` ( * ` ( F ` x ) ) ) = ( F ` x ) )
41	40	mpteq2dva 4018	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) = ( x e. X |-> ( F ` x ) ) )
42	39	cjcld 10712	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( * ` ( F ` x ) ) e. CC )
43		simpl 108	. . . . . . . . . . . . 13 |- ( ( F : X --> CC /\ X C_ RR ) -> F : X --> CC )
44	43	feqmptd 5474	. . . . . . . . . . . 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 5474	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> * = ( y e. CC |-> ( * ` y ) ) )
47		fveq2 5421	. . . . . . . . . . . 12 |- ( y = ( F ` x ) -> ( * ` y ) = ( * ` ( F ` x ) ) )
48	39, 44, 46, 47	fmptco 5586	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) = ( x e. X |-> ( * ` ( F ` x ) ) ) )
49		fveq2 5421	. . . . . . . . . . 11 |- ( y = ( * ` ( F ` x ) ) -> ( * ` y ) = ( * ` ( * ` ( F ` x ) ) ) )
50	42, 48, 46, 49	fmptco 5586	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) )
51	41, 50, 44	3eqtr4d 2182	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = F )
52	51	oveq2d 5790	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. ( * o. F ) ) ) = ( RR _D F ) )
53	52	dmeqd 4741	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. ( * o. F ) ) ) = dom ( RR _D F ) )
54	37, 53	sseqtrd 3135	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D F ) )
55		ffdm 5293	. . . . . . . . . . . . 13 |- ( F : X --> CC -> ( F : dom F --> CC /\ dom F C_ X ) )
56	55	simpld 111	. . . . . . . . . . . 12 |- ( F : X --> CC -> F : dom F --> CC )
57	56	adantr 274	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> F : dom F --> CC )
58		fdm 5278	. . . . . . . . . . . . 13 |- ( F : X --> CC -> dom F = X )
59	58	adantr 274	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> dom F = X )
60	59, 13	eqsstrd 3133	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> dom F C_ RR )
61	15, 16	elpm2 6574	. . . . . . . . . . 11 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
62	57, 60, 61	sylanbrc 413	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> F e. ( CC ^pm RR ) )
63		dvfpm 12827	. . . . . . . . . 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	ffvelrnda 5555	. . . . . . . 8 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
66	65	cjcld 10712	. . . . . . 7 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( * ` ( ( RR _D F ) ` x ) ) e. CC )
67		breldmg 4745	. . . . . . 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 1320	. . . . . 6 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x e. dom ( RR _D ( * o. F ) ) )
69	54, 68	eqelssd 3116	. . . . 5 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) = dom ( RR _D F ) )
70	69	feq2d 5260	. . . 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 146	. . 3 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) : dom ( RR _D F ) --> CC )
72	71	feqmptd 5474	. 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 5474	. . 3 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D F ) = ( x e. dom ( RR _D F ) |-> ( ( RR _D F ) ` x ) ) )
74		fveq2 5421	. . 3 |- ( y = ( ( RR _D F ) ` x ) -> ( * ` y ) = ( * ` ( ( RR _D F ) ` x ) ) )
75	65, 73, 46, 74	fmptco 5586	. 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 2182	1 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   C_ wss 3071   class class class wbr 3929   |-> cmpt 3989   dom cdm 4539   o. ccom 4543   Fun wfun 5117   -->wf 5119   ` cfv 5123  (class class class)co 5774   ^pm cpm 6543   CCcc 7618   RRcr 7619   *ccj 10611   _D cdv 12793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-pm 6545  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-ioo 9675  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-rest 12122  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-met 12158  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210  df-ntr 12265  df-cn 12357  df-cnp 12358  df-cncf 12727  df-limced 12794  df-dvap 12795

This theorem is referenced by:  dvfre  12843