Theorem dvcj 14888

Description: The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 14887. (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 14887	. . . 4 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) )
5		cjf 10994	. . . . . . . . . . . 12 |- * : CC --> CC
6		fco 5420	. . . . . . . . . . . 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 5411	. . . . . . . . . . . 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 5392	. . . . . . . . . 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 3216	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( * o. F ) C_ RR )
15		cnex 7998	. . . . . . . . . 10 |- CC e. _V
16		reex 8008	. . . . . . . . . 10 |- RR e. _V
17	15, 16	elpm2 6736	. . . . . . . . 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 14868	. . . . . . . 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 5408	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> Fun ( RR _D ( * o. F ) ) )
22		funbrfv 5596	. . . . . 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 4120	. 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 2763	. . . . . . . . . 10 |- x e. _V
28	20	ffvelcdmda 5694	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( ( RR _D ( * o. F ) ) ` x ) e. CC )
29	28	cjcld 11087	. . . . . . . . . 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 14887	. . . . . . . . . 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 4869	. . . . . . . . . 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 1353	. . . . . . . . 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 3186	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D ( * o. ( * o. F ) ) ) )
38		ffvelcdm 5692	. . . . . . . . . . . . 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 11090	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( * ` ( * ` ( F ` x ) ) ) = ( F ` x ) )
41	40	mpteq2dva 4120	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) = ( x e. X |-> ( F ` x ) ) )
42	39	cjcld 11087	. . . . . . . . . . 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 5611	. . . . . . . . . . . 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 5611	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> * = ( y e. CC |-> ( * ` y ) ) )
47		fveq2 5555	. . . . . . . . . . . 12 |- ( y = ( F ` x ) -> ( * ` y ) = ( * ` ( F ` x ) ) )
48	39, 44, 46, 47	fmptco 5725	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) = ( x e. X |-> ( * ` ( F ` x ) ) ) )
49		fveq2 5555	. . . . . . . . . . 11 |- ( y = ( * ` ( F ` x ) ) -> ( * ` y ) = ( * ` ( * ` ( F ` x ) ) ) )
50	42, 48, 46, 49	fmptco 5725	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) )
51	41, 50, 44	3eqtr4d 2236	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = F )
52	51	oveq2d 5935	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. ( * o. F ) ) ) = ( RR _D F ) )
53	52	dmeqd 4865	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. ( * o. F ) ) ) = dom ( RR _D F ) )
54	37, 53	sseqtrd 3218	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D F ) )
55		ffdm 5425	. . . . . . . . . . . . 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 5410	. . . . . . . . . . . . 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 3216	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> dom F C_ RR )
61	15, 16	elpm2 6736	. . . . . . . . . . 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 14868	. . . . . . . . . 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 5694	. . . . . . . 8 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
66	65	cjcld 11087	. . . . . . 7 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( * ` ( ( RR _D F ) ` x ) ) e. CC )
67		breldmg 4869	. . . . . . 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 1353	. . . . . 6 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x e. dom ( RR _D ( * o. F ) ) )
69	54, 68	eqelssd 3199	. . . . 5 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) = dom ( RR _D F ) )
70	69	feq2d 5392	. . . 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 5611	. 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 5611	. . 3 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D F ) = ( x e. dom ( RR _D F ) |-> ( ( RR _D F ) ` x ) ) )
74		fveq2 5555	. . 3 |- ( y = ( ( RR _D F ) ` x ) -> ( * ` y ) = ( * ` ( ( RR _D F ) ` x ) ) )
75	65, 73, 46, 74	fmptco 5725	. 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 2236	1 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3154   class class class wbr 4030   |-> cmpt 4091   dom cdm 4660   o. ccom 4664   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5919   ^pm cpm 6705   CCcc 7872   RRcr 7873   *ccj 10986   _D cdv 14834

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-map 6706  df-pm 6707  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-xneg 9841  df-xadd 9842  df-ioo 9961  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-rest 12855  df-topgen 12874  df-psmet 14042  df-xmet 14043  df-met 14044  df-bl 14045  df-mopn 14046  df-top 14177  df-topon 14190  df-bases 14222  df-ntr 14275  df-cn 14367  df-cnp 14368  df-cncf 14750  df-limced 14835  df-dvap 14836

This theorem is referenced by:  dvfre  14889  dvmptcjx  14903