Theorem dvfre 15030

Description: The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)

Assertion

Ref	Expression
dvfre	|- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )

Proof of Theorem dvfre

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-resscn 7988	. . . . . . 7 |- RR C_ CC
2		fss 5422	. . . . . . 7 |- ( ( F : A --> RR /\ RR C_ CC ) -> F : A --> CC )
3	1, 2	mpan2 425	. . . . . 6 |- ( F : A --> RR -> F : A --> CC )
4	3	adantr 276	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> F : A --> CC )
5		ffdm 5431	. . . . . 6 |- ( F : A --> CC -> ( F : dom F --> CC /\ dom F C_ A ) )
6	5	simpld 112	. . . . 5 |- ( F : A --> CC -> F : dom F --> CC )
7	4, 6	syl 14	. . . 4 |- ( ( F : A --> RR /\ A C_ RR ) -> F : dom F --> CC )
8		simpl 109	. . . . . 6 |- ( ( F : A --> RR /\ A C_ RR ) -> F : A --> RR )
9	8	fdmd 5417	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> dom F = A )
10		simpr 110	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> A C_ RR )
11	9, 10	eqsstrd 3220	. . . 4 |- ( ( F : A --> RR /\ A C_ RR ) -> dom F C_ RR )
12		cnex 8020	. . . . 5 |- CC e. _V
13		reex 8030	. . . . 5 |- RR e. _V
14	12, 13	elpm2 6748	. . . 4 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
15	7, 11, 14	sylanbrc 417	. . 3 |- ( ( F : A --> RR /\ A C_ RR ) -> F e. ( CC ^pm RR ) )
16		dvfpm 15009	. . 3 |- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
17		ffn 5410	. . 3 |- ( ( RR _D F ) : dom ( RR _D F ) --> CC -> ( RR _D F ) Fn dom ( RR _D F ) )
18	15, 16, 17	3syl 17	. 2 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D F ) Fn dom ( RR _D F ) )
19	15, 16	syl 14	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
20	19	ffvelcdmda 5700	. . . 4 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
21		fvco3 5635	. . . . . 6 |- ( ( ( RR _D F ) : dom ( RR _D F ) --> CC /\ x e. dom ( RR _D F ) ) -> ( ( * o. ( RR _D F ) ) ` x ) = ( * ` ( ( RR _D F ) ` x ) ) )
22	19, 21	sylan 283	. . . . 5 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( * o. ( RR _D F ) ) ` x ) = ( * ` ( ( RR _D F ) ` x ) ) )
23		dvcj 15029	. . . . . . . . 9 |- ( ( F : A --> CC /\ A C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )
24	3, 23	sylan 283	. . . . . . . 8 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )
25		ffvelcdm 5698	. . . . . . . . . . . . 13 |- ( ( F : A --> RR /\ y e. A ) -> ( F ` y ) e. RR )
26	25	adantlr 477	. . . . . . . . . . . 12 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( F ` y ) e. RR )
27	26	cjred 11153	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( * ` ( F ` y ) ) = ( F ` y ) )
28	27	mpteq2dva 4124	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( y e. A |-> ( * ` ( F ` y ) ) ) = ( y e. A |-> ( F ` y ) ) )
29	26	recnd 8072	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( F ` y ) e. CC )
30	8	feqmptd 5617	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> F = ( y e. A |-> ( F ` y ) ) )
31		cjf 11029	. . . . . . . . . . . . 13 |- * : CC --> CC
32	31	a1i 9	. . . . . . . . . . . 12 |- ( ( F : A --> RR /\ A C_ RR ) -> * : CC --> CC )
33	32	feqmptd 5617	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> * = ( z e. CC |-> ( * ` z ) ) )
34		fveq2 5561	. . . . . . . . . . 11 |- ( z = ( F ` y ) -> ( * ` z ) = ( * ` ( F ` y ) ) )
35	29, 30, 33, 34	fmptco 5731	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = ( y e. A |-> ( * ` ( F ` y ) ) ) )
36	28, 35, 30	3eqtr4d 2239	. . . . . . . . 9 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = F )
37	36	oveq2d 5941	. . . . . . . 8 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D ( * o. F ) ) = ( RR _D F ) )
38	24, 37	eqtr3d 2231	. . . . . . 7 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. ( RR _D F ) ) = ( RR _D F ) )
39	38	fveq1d 5563	. . . . . 6 |- ( ( F : A --> RR /\ A C_ RR ) -> ( ( * o. ( RR _D F ) ) ` x ) = ( ( RR _D F ) ` x ) )
40	39	adantr 276	. . . . 5 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( * o. ( RR _D F ) ) ` x ) = ( ( RR _D F ) ` x ) )
41	22, 40	eqtr3d 2231	. . . 4 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( * ` ( ( RR _D F ) ` x ) ) = ( ( RR _D F ) ` x ) )
42	20, 41	cjrebd 11128	. . 3 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. RR )
43	42	ralrimiva 2570	. 2 |- ( ( F : A --> RR /\ A C_ RR ) -> A. x e. dom ( RR _D F ) ( ( RR _D F ) ` x ) e. RR )
44		ffnfv 5723	. 2 |- ( ( RR _D F ) : dom ( RR _D F ) --> RR <-> ( ( RR _D F ) Fn dom ( RR _D F ) /\ A. x e. dom ( RR _D F ) ( ( RR _D F ) ` x ) e. RR ) )
45	18, 43, 44	sylanbrc 417	1 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   |-> cmpt 4095   dom cdm 4664   o. ccom 4668   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   ^pm cpm 6717   CCcc 7894   RRcr 7895   *ccj 11021   _D cdv 14975

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

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 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-pm 6719  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-xneg 9864  df-xadd 9865  df-ioo 9984  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-rest 12943  df-topgen 12962  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178  df-mopn 14179  df-top 14318  df-topon 14331  df-bases 14363  df-ntr 14416  df-cn 14508  df-cnp 14509  df-cncf 14891  df-limced 14976  df-dvap 14977

This theorem is referenced by: (None)