Theorem dvfre 15436

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 8124	. . . . . . 7 |- RR C_ CC
2		fss 5494	. . . . . . 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 5505	. . . . . 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 5489	. . . . 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 3263	. . . 4 |- ( ( F : A --> RR /\ A C_ RR ) -> dom F C_ RR )
12		cnex 8156	. . . . 5 |- CC e. _V
13		reex 8166	. . . . 5 |- RR e. _V
14	12, 13	elpm2 6849	. . . 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 15415	. . 3 |- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
17		ffn 5482	. . 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 5782	. . . 4 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. CC )
21		fvco3 5717	. . . . . 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 15435	. . . . . . . . 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 5780	. . . . . . . . . . . . 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 11532	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( * ` ( F ` y ) ) = ( F ` y ) )
28	27	mpteq2dva 4179	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( y e. A |-> ( * ` ( F ` y ) ) ) = ( y e. A |-> ( F ` y ) ) )
29	26	recnd 8208	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ y e. A ) -> ( F ` y ) e. CC )
30	8	feqmptd 5699	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> F = ( y e. A |-> ( F ` y ) ) )
31		cjf 11408	. . . . . . . . . . . . 13 |- * : CC --> CC
32	31	a1i 9	. . . . . . . . . . . 12 |- ( ( F : A --> RR /\ A C_ RR ) -> * : CC --> CC )
33	32	feqmptd 5699	. . . . . . . . . . 11 |- ( ( F : A --> RR /\ A C_ RR ) -> * = ( z e. CC |-> ( * ` z ) ) )
34		fveq2 5639	. . . . . . . . . . 11 |- ( z = ( F ` y ) -> ( * ` z ) = ( * ` ( F ` y ) ) )
35	29, 30, 33, 34	fmptco 5813	. . . . . . . . . 10 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = ( y e. A |-> ( * ` ( F ` y ) ) ) )
36	28, 35, 30	3eqtr4d 2274	. . . . . . . . 9 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. F ) = F )
37	36	oveq2d 6034	. . . . . . . 8 |- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D ( * o. F ) ) = ( RR _D F ) )
38	24, 37	eqtr3d 2266	. . . . . . 7 |- ( ( F : A --> RR /\ A C_ RR ) -> ( * o. ( RR _D F ) ) = ( RR _D F ) )
39	38	fveq1d 5641	. . . . . 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 2266	. . . 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 11507	. . 3 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ x e. dom ( RR _D F ) ) -> ( ( RR _D F ) ` x ) e. RR )
43	42	ralrimiva 2605	. 2 |- ( ( F : A --> RR /\ A C_ RR ) -> A. x e. dom ( RR _D F ) ( ( RR _D F ) ` x ) e. RR )
44		ffnfv 5805	. 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 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   |-> cmpt 4150   dom cdm 4725   o. ccom 4729   Fn wfn 5321   -->wf 5322   ` cfv 5326  (class class class)co 6018   ^pm cpm 6818   CCcc 8030   RRcr 8031   *ccj 11400   _D cdv 15381

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-pm 6820  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-ioo 10127  df-seqfrec 10710  df-exp 10801  df-cj 11403  df-re 11404  df-im 11405  df-rsqrt 11559  df-abs 11560  df-rest 13325  df-topgen 13344  df-psmet 14559  df-xmet 14560  df-met 14561  df-bl 14562  df-mopn 14563  df-top 14724  df-topon 14737  df-bases 14769  df-ntr 14822  df-cn 14914  df-cnp 14915  df-cncf 15297  df-limced 15382  df-dvap 15383

This theorem is referenced by: (None)