Theorem dvcj 14945

Description: The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 14944. (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 14944	. . . 4 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> x ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` x ) ) )
5		cjf 11012	. . . . . . . . . . . 12 |- * : CC --> CC
6		fco 5423	. . . . . . . . . . . 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 5414	. . . . . . . . . . . 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 5395	. . . . . . . . . 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 3219	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( * o. F ) C_ RR )
15		cnex 8003	. . . . . . . . . 10 |- CC e. _V
16		reex 8013	. . . . . . . . . 10 |- RR e. _V
17	15, 16	elpm2 6739	. . . . . . . . 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 14925	. . . . . . . 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 5411	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> Fun ( RR _D ( * o. F ) ) )
22		funbrfv 5599	. . . . . 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 4123	. 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 2766	. . . . . . . . . 10 |- x e. _V
28	20	ffvelcdmda 5697	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D ( * o. F ) ) ) -> ( ( RR _D ( * o. F ) ) ` x ) e. CC )
29	28	cjcld 11105	. . . . . . . . . 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 14944	. . . . . . . . . 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 4872	. . . . . . . . . 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 3189	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D ( * o. ( * o. F ) ) ) )
38		ffvelcdm 5695	. . . . . . . . . . . . 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 11108	. . . . . . . . . . 11 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. X ) -> ( * ` ( * ` ( F ` x ) ) ) = ( F ` x ) )
41	40	mpteq2dva 4123	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) = ( x e. X |-> ( F ` x ) ) )
42	39	cjcld 11105	. . . . . . . . . . 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 5614	. . . . . . . . . . . 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 5614	. . . . . . . . . . . 12 |- ( ( F : X --> CC /\ X C_ RR ) -> * = ( y e. CC |-> ( * ` y ) ) )
47		fveq2 5558	. . . . . . . . . . . 12 |- ( y = ( F ` x ) -> ( * ` y ) = ( * ` ( F ` x ) ) )
48	39, 44, 46, 47	fmptco 5728	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. F ) = ( x e. X |-> ( * ` ( F ` x ) ) ) )
49		fveq2 5558	. . . . . . . . . . 11 |- ( y = ( * ` ( F ` x ) ) -> ( * ` y ) = ( * ` ( * ` ( F ` x ) ) ) )
50	42, 48, 46, 49	fmptco 5728	. . . . . . . . . 10 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = ( x e. X |-> ( * ` ( * ` ( F ` x ) ) ) ) )
51	41, 50, 44	3eqtr4d 2239	. . . . . . . . 9 |- ( ( F : X --> CC /\ X C_ RR ) -> ( * o. ( * o. F ) ) = F )
52	51	oveq2d 5938	. . . . . . . 8 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. ( * o. F ) ) ) = ( RR _D F ) )
53	52	dmeqd 4868	. . . . . . 7 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. ( * o. F ) ) ) = dom ( RR _D F ) )
54	37, 53	sseqtrd 3221	. . . . . 6 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) C_ dom ( RR _D F ) )
55		ffdm 5428	. . . . . . . . . . . . 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 5413	. . . . . . . . . . . . 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 3219	. . . . . . . . . . 11 |- ( ( F : X --> CC /\ X C_ RR ) -> dom F C_ RR )
61	15, 16	elpm2 6739	. . . . . . . . . . 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 14925	. . . . . . . . . 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 5697	. . . . . . . 8 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
66	65	cjcld 11105	. . . . . . 7 |- ( ( ( F : X --> CC /\ X C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( * ` ( ( RR _D F ) ` x ) ) e. CC )
67		breldmg 4872	. . . . . . 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 3202	. . . . 5 |- ( ( F : X --> CC /\ X C_ RR ) -> dom ( RR _D ( * o. F ) ) = dom ( RR _D F ) )
70	69	feq2d 5395	. . . 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 5614	. 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 5614	. . 3 |- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D F ) = ( x e. dom ( RR _D F ) |-> ( ( RR _D F ) ` x ) ) )
74		fveq2 5558	. . 3 |- ( y = ( ( RR _D F ) ` x ) -> ( * ` y ) = ( * ` ( ( RR _D F ) ` x ) ) )
75	65, 73, 46, 74	fmptco 5728	. 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 2239	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 2167   _Vcvv 2763   C_ wss 3157   class class class wbr 4033   |-> cmpt 4094   dom cdm 4663   o. ccom 4667   Fun wfun 5252   -->wf 5254   ` cfv 5258  (class class class)co 5922   ^pm cpm 6708   CCcc 7877   RRcr 7878   *ccj 11004   _D cdv 14891

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-map 6709  df-pm 6710  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-ioo 9967  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-rest 12912  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102  df-mopn 14103  df-top 14234  df-topon 14247  df-bases 14279  df-ntr 14332  df-cn 14424  df-cnp 14425  df-cncf 14807  df-limced 14892  df-dvap 14893

This theorem is referenced by:  dvfre  14946  dvmptcjx  14960