Theorem dviaddf 13463

Description: The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Hypotheses

Ref	Expression
dvaddf.s	|- ( ph -> S e. { RR , CC } )
dviaddf.x	|- ( ph -> X C_ S )
dvaddf.f	|- ( ph -> F : X --> CC )
dvaddf.g	|- ( ph -> G : X --> CC )
dvaddf.df	|- ( ph -> dom ( S _D F ) = X )
dvaddf.dg	|- ( ph -> dom ( S _D G ) = X )

Assertion

Ref	Expression
dviaddf	|- ( ph -> ( S _D ( F oF + G ) ) = ( ( S _D F ) oF + ( S _D G ) ) )

Proof of Theorem dviaddf

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

Step	Hyp	Ref	Expression
1		addcl 7899	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
2	1	adantl 275	. . 3 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
3		dvaddf.s	. . . . 5 |- ( ph -> S e. { RR , CC } )
4		cnex 7898	. . . . . . 7 |- CC e. _V
5	4	a1i 9	. . . . . 6 |- ( ph -> CC e. _V )
6		dvaddf.f	. . . . . 6 |- ( ph -> F : X --> CC )
7		dviaddf.x	. . . . . 6 |- ( ph -> X C_ S )
8		elpm2r 6644	. . . . . 6 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : X --> CC /\ X C_ S ) ) -> F e. ( CC ^pm S ) )
9	5, 3, 6, 7, 8	syl22anc 1234	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
10		dvfgg 13451	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
11	3, 9, 10	syl2anc 409	. . . 4 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
12		dvaddf.df	. . . . 5 |- ( ph -> dom ( S _D F ) = X )
13	12	feq2d 5335	. . . 4 |- ( ph -> ( ( S _D F ) : dom ( S _D F ) --> CC <-> ( S _D F ) : X --> CC ) )
14	11, 13	mpbid 146	. . 3 |- ( ph -> ( S _D F ) : X --> CC )
15		dvaddf.g	. . . . . 6 |- ( ph -> G : X --> CC )
16		elpm2r 6644	. . . . . 6 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( G : X --> CC /\ X C_ S ) ) -> G e. ( CC ^pm S ) )
17	5, 3, 15, 7, 16	syl22anc 1234	. . . . 5 |- ( ph -> G e. ( CC ^pm S ) )
18		dvfgg 13451	. . . . 5 |- ( ( S e. { RR , CC } /\ G e. ( CC ^pm S ) ) -> ( S _D G ) : dom ( S _D G ) --> CC )
19	3, 17, 18	syl2anc 409	. . . 4 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
20		dvaddf.dg	. . . . 5 |- ( ph -> dom ( S _D G ) = X )
21	20	feq2d 5335	. . . 4 |- ( ph -> ( ( S _D G ) : dom ( S _D G ) --> CC <-> ( S _D G ) : X --> CC ) )
22	19, 21	mpbid 146	. . 3 |- ( ph -> ( S _D G ) : X --> CC )
23	3, 7	ssexd 4129	. . 3 |- ( ph -> X e. _V )
24		inidm 3336	. . 3 |- ( X i^i X ) = X
25	2, 6, 15, 23, 23, 24	off 6073	. . . . . 6 |- ( ph -> ( F oF + G ) : X --> CC )
26		elpm2r 6644	. . . . . 6 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( ( F oF + G ) : X --> CC /\ X C_ S ) ) -> ( F oF + G ) e. ( CC ^pm S ) )
27	5, 3, 25, 7, 26	syl22anc 1234	. . . . 5 |- ( ph -> ( F oF + G ) e. ( CC ^pm S ) )
28		dvfgg 13451	. . . . 5 |- ( ( S e. { RR , CC } /\ ( F oF + G ) e. ( CC ^pm S ) ) -> ( S _D ( F oF + G ) ) : dom ( S _D ( F oF + G ) ) --> CC )
29	3, 27, 28	syl2anc 409	. . . 4 |- ( ph -> ( S _D ( F oF + G ) ) : dom ( S _D ( F oF + G ) ) --> CC )
30		recnprss 13450	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
31	3, 30	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
32	31, 25, 7	dvbss 13448	. . . . . 6 |- ( ph -> dom ( S _D ( F oF + G ) ) C_ X )
33		reldvg 13442	. . . . . . . . 9 |- ( ( S C_ CC /\ ( F oF + G ) e. ( CC ^pm S ) ) -> Rel ( S _D ( F oF + G ) ) )
34	31, 27, 33	syl2anc 409	. . . . . . . 8 |- ( ph -> Rel ( S _D ( F oF + G ) ) )
35	34	adantr 274	. . . . . . 7 |- ( ( ph /\ x e. X ) -> Rel ( S _D ( F oF + G ) ) )
36	6	adantr 274	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> F : X --> CC )
37	7	adantr 274	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> X C_ S )
38	15	adantr 274	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> G : X --> CC )
39	31	adantr 274	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> S C_ CC )
40	12	eleq2d 2240	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D F ) <-> x e. X ) )
41	40	biimpar 295	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D F ) )
42		ffun 5350	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
43		funfvbrb 5609	. . . . . . . . . . 11 |- ( Fun ( S _D F ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
44	11, 42, 43	3syl 17	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
45	44	adantr 274	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
46	41, 45	mpbid 146	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D F ) ( ( S _D F ) ` x ) )
47	20	eleq2d 2240	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D G ) <-> x e. X ) )
48	47	biimpar 295	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D G ) )
49		ffun 5350	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
50		funfvbrb 5609	. . . . . . . . . . 11 |- ( Fun ( S _D G ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
51	19, 49, 50	3syl 17	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
52	51	adantr 274	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
53	48, 52	mpbid 146	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D G ) ( ( S _D G ) ` x ) )
54		eqid 2170	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
55	36, 37, 38, 39, 46, 53, 54	dvaddxxbr 13459	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D ( F oF + G ) ) ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
56		releldm 4846	. . . . . . 7 |- ( ( Rel ( S _D ( F oF + G ) ) /\ x ( S _D ( F oF + G ) ) ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) ) -> x e. dom ( S _D ( F oF + G ) ) )
57	35, 55, 56	syl2anc 409	. . . . . 6 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D ( F oF + G ) ) )
58	32, 57	eqelssd 3166	. . . . 5 |- ( ph -> dom ( S _D ( F oF + G ) ) = X )
59	58	feq2d 5335	. . . 4 |- ( ph -> ( ( S _D ( F oF + G ) ) : dom ( S _D ( F oF + G ) ) --> CC <-> ( S _D ( F oF + G ) ) : X --> CC ) )
60	29, 59	mpbid 146	. . 3 |- ( ph -> ( S _D ( F oF + G ) ) : X --> CC )
61		eqidd 2171	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( S _D F ) ` x ) )
62		eqidd 2171	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D G ) ` x ) = ( ( S _D G ) ` x ) )
63	3	adantr 274	. . . . 5 |- ( ( ph /\ x e. X ) -> S e. { RR , CC } )
64	36, 37, 38, 63, 41, 48	dvaddxx 13461	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D ( F oF + G ) ) ` x ) = ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
65	64	eqcomd 2176	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) = ( ( S _D ( F oF + G ) ) ` x ) )
66	2, 14, 22, 23, 23, 24, 60, 61, 62, 65	offeq 6074	. 2 |- ( ph -> ( ( S _D F ) oF + ( S _D G ) ) = ( S _D ( F oF + G ) ) )
67	66	eqcomd 2176	1 |- ( ph -> ( S _D ( F oF + G ) ) = ( ( S _D F ) oF + ( S _D G ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   {cpr 3584   class class class wbr 3989   dom cdm 4611   o. ccom 4615   Rel wrel 4616   Fun wfun 5192   -->wf 5194   ` cfv 5198  (class class class)co 5853   oFcof 6059   ^pm cpm 6627   CCcc 7772   RRcr 7773   + caddc 7777   - cmin 8090   abscabs 10961   MetOpencmopn 12779   _D cdv 13418

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894  ax-addf 7896

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-pm 6629  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-rest 12581  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-met 12783  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835  df-ntr 12890  df-cn 12982  df-cnp 12983  df-tx 13047  df-limced 13419  df-dvap 13420

This theorem is referenced by:  dvmptaddx  13475