Theorem dvcj 15296

Description: The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 15295. (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 15295	. . . 4 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) )
5		cjf 11273	. . . . . . . . . . . 12 |- * : CC --> CC
6		fco 5461	. . . . . . . . . . . 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 5452	. . . . . . . . . . . 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 5433	. . . . . . . . . 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 3237	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( * o. F ) C_ RR )
15		cnex 8084	. . . . . . . . . 10 |- CC e. _V
16		reex 8094	. . . . . . . . . 10 |- RR e. _V
17	15, 16	elpm2 6790	. . . . . . . . 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 15276	. . . . . . . 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 5449	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> Fun ( RR _D ( * o. F ) ) )
22		funbrfv 5640	. . . . . 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 4150	. 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 2779	. . . . . . . . . 10 |- x e. _V
28	20	ffvelcdmda 5738	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( ( RR _D ( * o. F ) ) ` x ) e. CC )
29	28	cjcld 11366	. . . . . . . . . 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 15295	. . . . . . . . . 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 4903	. . . . . . . . . 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 1355	. . . . . . . . 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 3207	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D ( * o. ( * o. F ) ) ) )
38		ffvelcdm 5736	. . . . . . . . . . . . 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 11369	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( * ` ( * ` ( F ` x ) ) ) = ( F ` x ) )
41	40	mpteq2dva 4150	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) = ( x e. X |-> ( F ` x ) ) )
42	39	cjcld 11366	. . . . . . . . . . 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 5655	. . . . . . . . . . . 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 5655	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> * = ( y e. CC |-> ( * ` y ) ) )
47		fveq2 5599	. . . . . . . . . . . 12 |- ( y = ( F ` x ) -> ( * ` y ) = ( * ` ( F ` x ) ) )
48	39, 44, 46, 47	fmptco 5769	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) = ( x e. X |-> ( * ` ( F ` x ) ) ) )
49		fveq2 5599	. . . . . . . . . . 11 |- ( y = ( * ` ( F ` x ) ) -> ( * ` y ) = ( * ` ( * ` ( F ` x ) ) ) )
50	42, 48, 46, 49	fmptco 5769	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) )
51	41, 50, 44	3eqtr4d 2250	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = F )
52	51	oveq2d 5983	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. ( * o. F ) ) ) = ( RR _D F ) )
53	52	dmeqd 4899	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. ( * o. F ) ) ) = dom ( RR _D F ) )
54	37, 53	sseqtrd 3239	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D F ) )
55		ffdm 5466	. . . . . . . . . . . . 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 5451	. . . . . . . . . . . . 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 3237	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> dom F C_ RR )
61	15, 16	elpm2 6790	. . . . . . . . . . 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 15276	. . . . . . . . . 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 5738	. . . . . . . 8 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
66	65	cjcld 11366	. . . . . . 7 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( * ` ( ( RR _D F ) ` x ) ) e. CC )
67		breldmg 4903	. . . . . . 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 1355	. . . . . 6 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x e. dom ( RR _D ( * o. F ) ) )
69	54, 68	eqelssd 3220	. . . . 5 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) = dom ( RR _D F ) )
70	69	feq2d 5433	. . . 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 5655	. 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 5655	. . 3 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D F ) = ( x e. dom ( RR _D F ) |-> ( ( RR _D F ) ` x ) ) )
74		fveq2 5599	. . 3 |- ( y = ( ( RR _D F ) ` x ) -> ( * ` y ) = ( * ` ( ( RR _D F ) ` x ) ) )
75	65, 73, 46, 74	fmptco 5769	. 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 2250	1 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   class class class wbr 4059   |-> cmpt 4121   dom cdm 4693   o. ccom 4697   Fun wfun 5284   -->wf 5286   ` cfv 5290  (class class class)co 5967   ^pm cpm 6759   CCcc 7958   RRcr 7959   *ccj 11265   _D cdv 15242

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-map 6760  df-pm 6761  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-xneg 9929  df-xadd 9930  df-ioo 10049  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-rest 13188  df-topgen 13207  df-psmet 14420  df-xmet 14421  df-met 14422  df-bl 14423  df-mopn 14424  df-top 14585  df-topon 14598  df-bases 14630  df-ntr 14683  df-cn 14775  df-cnp 14776  df-cncf 15158  df-limced 15243  df-dvap 15244

This theorem is referenced by:  dvfre  15297  dvmptcjx  15311