Theorem dvfre 13074

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 7824	. . . . . . 7 |- RR C_ CC
2		fss 5331	. . . . . . 7 |- ( ( F : A --> RR /\ RR C_ CC ) -> F : A --> CC )
3	1, 2	mpan2 422	. . . . . 6 |- ( F : A --> RR -> F : A --> CC )
4	3	adantr 274	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> F : A --> CC )
5		ffdm 5340	. . . . . 6 |- ( F : A --> CC -> ( F : dom F --> CC /\ dom F C_ A ) )
6	5	simpld 111	. . . . 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 108	. . . . . 6 |- ( ( F : A --> RR /\ A C_ RR ) -> F : A --> RR )
9	8	fdmd 5326	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> dom F = A )
10		simpr 109	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> A C_ RR )
11	9, 10	eqsstrd 3164	. . . 4 |- ( ( F : A --> RR /\ A C_ RR ) -> dom F C_ RR )
12		cnex 7856	. . . . 5 |- CC e. _V
13		reex 7866	. . . . 5 |- RR e. _V
14	12, 13	elpm2 6625	. . . 4 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
15	7, 11, 14	sylanbrc 414	. . 3 |- ( ( F : A --> RR /\ A C_ RR ) -> F e. ( CC ^pm RR ) )
16		dvfpm 13058	. . 3 |- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
17		ffn 5319	. . 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	ffvelrnda 5602	. . . 4 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
21		fvco3 5539	. . . . . 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 281	. . . . 5 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( * o. ( RR _D F ) ) ` x ) = ( * ` ( ( RR _D F ) ` x ) ) )
23		dvcj 13073	. . . . . . . . 9 |- ( ( F : A --> CC /\ A C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )
24	3, 23	sylan 281	. . . . . . . 8 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )
25		ffvelrn 5600	. . . . . . . . . . . . 13 |- ( ( F : A --> RR /\ y e. A ) -> ( F ` y ) e. RR )
26	25	adantlr 469	. . . . . . . . . . . 12 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( F ` y ) e. RR )
27	26	cjred 10871	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( * ` ( F ` y ) ) = ( F ` y ) )
28	27	mpteq2dva 4054	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( y e. A |-> ( * ` ( F ` y ) ) ) = ( y e. A |-> ( F ` y ) ) )
29	26	recnd 7906	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( F ` y ) e. CC )
30	8	feqmptd 5521	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> F = ( y e. A |-> ( F ` y ) ) )
31		cjf 10747	. . . . . . . . . . . . 13 |- * : CC --> CC
32	31	a1i 9	. . . . . . . . . . . 12 |- ( ( F : A --> RR /\ A C_ RR ) -> * : CC --> CC )
33	32	feqmptd 5521	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> * = ( z e. CC |-> ( * ` z ) ) )
34		fveq2 5468	. . . . . . . . . . 11 |- ( z = ( F ` y ) -> ( * ` z ) = ( * ` ( F ` y ) ) )
35	29, 30, 33, 34	fmptco 5633	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = ( y e. A |-> ( * ` ( F ` y ) ) ) )
36	28, 35, 30	3eqtr4d 2200	. . . . . . . . 9 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = F )
37	36	oveq2d 5840	. . . . . . . 8 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D ( * o. F ) ) = ( RR _D F ) )
38	24, 37	eqtr3d 2192	. . . . . . 7 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. ( RR _D F ) ) = ( RR _D F ) )
39	38	fveq1d 5470	. . . . . 6 |- ( ( F : A --> RR /\ A C_ RR ) -> ( ( * o. ( RR _D F ) ) ` x ) = ( ( RR _D F ) ` x ) )
40	39	adantr 274	. . . . 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 2192	. . . 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 10846	. . 3 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. RR )
43	42	ralrimiva 2530	. 2 |- ( ( F : A --> RR /\ A C_ RR ) -> A. x e. dom ( RR _D F ) ( ( RR _D F ) ` x ) e. RR )
44		ffnfv 5625	. 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 414	1 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   A.wral 2435   C_ wss 3102   |-> cmpt 4025   dom cdm 4586   o. ccom 4590   Fn wfn 5165   -->wf 5166   ` cfv 5170  (class class class)co 5824   ^pm cpm 6594   CCcc 7730   RRcr 7731   *ccj 10739   _D cdv 13024

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-map 6595  df-pm 6596  df-sup 6928  df-inf 6929  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-xneg 9679  df-xadd 9680  df-ioo 9796  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-rest 12353  df-topgen 12372  df-psmet 12387  df-xmet 12388  df-met 12389  df-bl 12390  df-mopn 12391  df-top 12396  df-topon 12409  df-bases 12441  df-ntr 12496  df-cn 12588  df-cnp 12589  df-cncf 12958  df-limced 13025  df-dvap 13026

This theorem is referenced by: (None)