Theorem dvfre 13314

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 7845	. . . . . . 7 |- RR C_ CC
2		fss 5349	. . . . . . 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 5358	. . . . . 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 5344	. . . . 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 3178	. . . 4 |- ( ( F : A --> RR /\ A C_ RR ) -> dom F C_ RR )
12		cnex 7877	. . . . 5 |- CC e. _V
13		reex 7887	. . . . 5 |- RR e. _V
14	12, 13	elpm2 6646	. . . 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 13298	. . 3 |- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
17		ffn 5337	. . 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 5620	. . . 4 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
21		fvco3 5557	. . . . . 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 13313	. . . . . . . . 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 5618	. . . . . . . . . . . . 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 10913	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( * ` ( F ` y ) ) = ( F ` y ) )
28	27	mpteq2dva 4072	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( y e. A |-> ( * ` ( F ` y ) ) ) = ( y e. A |-> ( F ` y ) ) )
29	26	recnd 7927	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( F ` y ) e. CC )
30	8	feqmptd 5539	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> F = ( y e. A |-> ( F ` y ) ) )
31		cjf 10789	. . . . . . . . . . . . 13 |- * : CC --> CC
32	31	a1i 9	. . . . . . . . . . . 12 |- ( ( F : A --> RR /\ A C_ RR ) -> * : CC --> CC )
33	32	feqmptd 5539	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> * = ( z e. CC |-> ( * ` z ) ) )
34		fveq2 5486	. . . . . . . . . . 11 |- ( z = ( F ` y ) -> ( * ` z ) = ( * ` ( F ` y ) ) )
35	29, 30, 33, 34	fmptco 5651	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = ( y e. A |-> ( * ` ( F ` y ) ) ) )
36	28, 35, 30	3eqtr4d 2208	. . . . . . . . 9 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = F )
37	36	oveq2d 5858	. . . . . . . 8 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D ( * o. F ) ) = ( RR _D F ) )
38	24, 37	eqtr3d 2200	. . . . . . 7 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. ( RR _D F ) ) = ( RR _D F ) )
39	38	fveq1d 5488	. . . . . 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 2200	. . . 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 10888	. . 3 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. RR )
43	42	ralrimiva 2539	. 2 |- ( ( F : A --> RR /\ A C_ RR ) -> A. x e. dom ( RR _D F ) ( ( RR _D F ) ` x ) e. RR )
44		ffnfv 5643	. 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 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   |-> cmpt 4043   dom cdm 4604   o. ccom 4608   Fn wfn 5183   -->wf 5184   ` cfv 5188  (class class class)co 5842   ^pm cpm 6615   CCcc 7751   RRcr 7752   *ccj 10781   _D cdv 13264

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-map 6616  df-pm 6617  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ioo 9828  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-cncf 13198  df-limced 13265  df-dvap 13266

This theorem is referenced by: (None)