Theorem dviaddf 15806

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 8304	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
2	1	adantl 277	. . 3 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
3		dvaddf.s	. . . . 5 |- ( ph -> S e. { RR , CC } )
4		cnex 8303	. . . . . . 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 6940	. . . . . 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 1279	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
10		dvfgg 15789	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
11	3, 9, 10	syl2anc 415	. . . 4 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
12		dvaddf.df	. . . . 5 |- ( ph -> dom ( S _D F ) = X )
13	12	feq2d 5521	. . . 4 |- ( ph -> ( ( S _D F ) : dom ( S _D F ) --> CC <-> ( S _D F ) : X --> CC ) )
14	11, 13	mpbid 147	. . 3 |- ( ph -> ( S _D F ) : X --> CC )
15		dvaddf.g	. . . . . 6 |- ( ph -> G : X --> CC )
16		elpm2r 6940	. . . . . 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 1279	. . . . 5 |- ( ph -> G e. ( CC ^pm S ) )
18		dvfgg 15789	. . . . 5 |- ( ( S e. { RR , CC } /\ G e. ( CC ^pm S ) ) -> ( S _D G ) : dom ( S _D G ) --> CC )
19	3, 17, 18	syl2anc 415	. . . 4 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
20		dvaddf.dg	. . . . 5 |- ( ph -> dom ( S _D G ) = X )
21	20	feq2d 5521	. . . 4 |- ( ph -> ( ( S _D G ) : dom ( S _D G ) --> CC <-> ( S _D G ) : X --> CC ) )
22	19, 21	mpbid 147	. . 3 |- ( ph -> ( S _D G ) : X --> CC )
23	3, 7	ssexd 4273	. . 3 |- ( ph -> X e. _V )
24		inidm 3440	. . 3 |- ( X i^i X ) = X
25	2, 6, 15, 23, 23, 24	off 6315	. . . . . 6 |- ( ph -> ( F oF + G ) : X --> CC )
26		elpm2r 6940	. . . . . 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 1279	. . . . 5 |- ( ph -> ( F oF + G ) e. ( CC ^pm S ) )
28		dvfgg 15789	. . . . 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 415	. . . 4 |- ( ph -> ( S _D ( F oF + G ) ) : dom ( S _D ( F oF + G ) ) --> CC )
30		recnprss 15788	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
31	3, 30	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
32	31, 25, 7	dvbss 15786	. . . . . 6 |- ( ph -> dom ( S _D ( F oF + G ) ) C_ X )
33		reldvg 15780	. . . . . . . . 9 |- ( ( S C_ CC /\ ( F oF + G ) e. ( CC ^pm S ) ) -> Rel ( S _D ( F oF + G ) ) )
34	31, 27, 33	syl2anc 415	. . . . . . . 8 |- ( ph -> Rel ( S _D ( F oF + G ) ) )
35	34	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. X ) -> Rel ( S _D ( F oF + G ) ) )
36	6	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> F : X --> CC )
37	7	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> X C_ S )
38	15	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> G : X --> CC )
39	31	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> S C_ CC )
40	12	eleq2d 2308	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D F ) <-> x e. X ) )
41	40	biimpar 297	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D F ) )
42		ffun 5536	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
43		funfvbrb 5822	. . . . . . . . . . 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 276	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
46	41, 45	mpbid 147	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D F ) ( ( S _D F ) ` x ) )
47	20	eleq2d 2308	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D G ) <-> x e. X ) )
48	47	biimpar 297	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D G ) )
49		ffun 5536	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
50		funfvbrb 5822	. . . . . . . . . . 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 276	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
53	48, 52	mpbid 147	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D G ) ( ( S _D G ) ` x ) )
54		eqid 2238	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
55	36, 37, 38, 39, 46, 53, 54	dvaddxxbr 15802	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D ( F oF + G ) ) ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
56		releldm 5017	. . . . . . 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 415	. . . . . 6 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D ( F oF + G ) ) )
58	32, 57	eqelssd 3267	. . . . 5 |- ( ph -> dom ( S _D ( F oF + G ) ) = X )
59	58	feq2d 5521	. . . 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 147	. . 3 |- ( ph -> ( S _D ( F oF + G ) ) : X --> CC )
61		eqidd 2239	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( S _D F ) ` x ) )
62		eqidd 2239	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D G ) ` x ) = ( ( S _D G ) ` x ) )
63	3	adantr 276	. . . . 5 |- ( ( ph /\ x e. X ) -> S e. { RR , CC } )
64	36, 37, 38, 63, 41, 48	dvaddxx 15804	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D ( F oF + G ) ) ` x ) = ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
65	64	eqcomd 2244	. . 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 6316	. 2 |- ( ph -> ( ( S _D F ) oF + ( S _D G ) ) = ( S _D ( F oF + G ) ) )
67	66	eqcomd 2244	1 |- ( ph -> ( S _D ( F oF + G ) ) = ( ( S _D F ) oF + ( S _D G ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {cpr 3710   class class class wbr 4130   dom cdm 4774   o. ccom 4778   Rel wrel 4779   Fun wfun 5371   -->wf 5373   ` cfv 5377  (class class class)co 6085   oFcof 6300   ^pm cpm 6923   CCcc 8177   RRcr 8178   + caddc 8182   - cmin 8497   abscabs 11763   MetOpencmopn 14878   _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  ax-addf 8301

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-of 6302  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-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-tx 15354  df-limced 15757  df-dvap 15758

This theorem is used by:  dvmptaddx  15820