Theorem dvfre 15811

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 8271	. . . . . . 7 |- RR C_ CC
2		fss 5546	. . . . . . 7 |- ( ( F : A --> RR /\ RR C_ CC ) -> F : A --> CC )
3	1, 2	mpan2 429	. . . . . 6 |- ( F : A --> RR -> F : A --> CC )
4	3	adantr 276	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> F : A --> CC )
5		ffdm 5558	. . . . . 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 5540	. . . . 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 3284	. . . 4 |- ( ( F : A --> RR /\ A C_ RR ) -> dom F C_ RR )
12		cnex 8303	. . . . 5 |- CC e. _V
13		reex 8313	. . . . 5 |- RR e. _V
14	12, 13	elpm2 6961	. . . 4 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
15	7, 11, 14	sylanbrc 421	. . 3 |- ( ( F : A --> RR /\ A C_ RR ) -> F e. ( CC ^pm RR ) )
16		dvfpm 15790	. . 3 |- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
17		ffn 5533	. . 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 5843	. . . 4 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
21		fvco3 5776	. . . . . 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 15810	. . . . . . . . 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 5841	. . . . . . . . . . . . 13 |- ( ( F : A --> RR /\ y e. A ) -> ( F ` y ) e. RR )
26	25	adantlr 481	. . . . . . . . . . . 12 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( F ` y ) e. RR )
27	26	cjred 11737	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( * ` ( F ` y ) ) = ( F ` y ) )
28	27	mpteq2dva 4221	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( y e. A |-> ( * ` ( F ` y ) ) ) = ( y e. A |-> ( F ` y ) ) )
29	26	recnd 8354	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( F ` y ) e. CC )
30	8	feqmptd 5756	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> F = ( y e. A |-> ( F ` y ) ) )
31		cjf 11612	. . . . . . . . . . . . 13 |- * : CC --> CC
32	31	a1i 9	. . . . . . . . . . . 12 |- ( ( F : A --> RR /\ A C_ RR ) -> * : CC --> CC )
33	32	feqmptd 5756	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> * = ( z e. CC |-> ( * ` z ) ) )
34		fveq2 5695	. . . . . . . . . . 11 |- ( z = ( F ` y ) -> ( * ` z ) = ( * ` ( F ` y ) ) )
35	29, 30, 33, 34	fmptco 5874	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = ( y e. A |-> ( * ` ( F ` y ) ) ) )
36	28, 35, 30	3eqtr4d 2281	. . . . . . . . 9 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = F )
37	36	oveq2d 6101	. . . . . . . 8 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D ( * o. F ) ) = ( RR _D F ) )
38	24, 37	eqtr3d 2273	. . . . . . 7 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. ( RR _D F ) ) = ( RR _D F ) )
39	38	fveq1d 5697	. . . . . 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 2273	. . . 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 11712	. . 3 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. RR )
43	42	ralrimiva 2623	. 2 |- ( ( F : A --> RR /\ A C_ RR ) -> A. x e. dom ( RR _D F ) ( ( RR _D F ) ` x ) e. RR )
44		ffnfv 5866	. 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 421	1 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   |-> cmpt 4192   dom cdm 4774   o. ccom 4778   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   ^pm cpm 6923   CCcc 8177   RRcr 8178   *ccj 11604   _D cdv 15756

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-pm 6925  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-ioo 10294  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-rest 13595  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883  df-mopn 14884  df-top 15099  df-topon 15112  df-bases 15144  df-ntr 15197  df-cn 15289  df-cnp 15290  df-cncf 15672  df-limced 15757  df-dvap 15758

This theorem is used by: (None)