Theorem dviaddf 14854

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 7997	. . . 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 7996	. . . . . . 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 6720	. . . . . 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 1250	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
10		dvfgg 14842	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
11	3, 9, 10	syl2anc 411	. . . 4 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
12		dvaddf.df	. . . . 5 |- ( ph -> dom ( S _D F ) = X )
13	12	feq2d 5391	. . . 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 6720	. . . . . 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 1250	. . . . 5 |- ( ph -> G e. ( CC ^pm S ) )
18		dvfgg 14842	. . . . 5 |- ( ( S e. { RR , CC } /\ G e. ( CC ^pm S ) ) -> ( S _D G ) : dom ( S _D G ) --> CC )
19	3, 17, 18	syl2anc 411	. . . 4 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
20		dvaddf.dg	. . . . 5 |- ( ph -> dom ( S _D G ) = X )
21	20	feq2d 5391	. . . 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 4169	. . 3 |- ( ph -> X e. _V )
24		inidm 3368	. . 3 |- ( X i^i X ) = X
25	2, 6, 15, 23, 23, 24	off 6143	. . . . . 6 |- ( ph -> ( F oF + G ) : X --> CC )
26		elpm2r 6720	. . . . . 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 1250	. . . . 5 |- ( ph -> ( F oF + G ) e. ( CC ^pm S ) )
28		dvfgg 14842	. . . . 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 411	. . . 4 |- ( ph -> ( S _D ( F oF + G ) ) : dom ( S _D ( F oF + G ) ) --> CC )
30		recnprss 14841	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
31	3, 30	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
32	31, 25, 7	dvbss 14839	. . . . . 6 |- ( ph -> dom ( S _D ( F oF + G ) ) C_ X )
33		reldvg 14833	. . . . . . . . 9 |- ( ( S C_ CC /\ ( F oF + G ) e. ( CC ^pm S ) ) -> Rel ( S _D ( F oF + G ) ) )
34	31, 27, 33	syl2anc 411	. . . . . . . 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 2263	. . . . . . . . . 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 5406	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
43		funfvbrb 5671	. . . . . . . . . . 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 2263	. . . . . . . . . 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 5406	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
50		funfvbrb 5671	. . . . . . . . . . 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 2193	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
55	36, 37, 38, 39, 46, 53, 54	dvaddxxbr 14850	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D ( F oF + G ) ) ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
56		releldm 4897	. . . . . . 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 411	. . . . . 6 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D ( F oF + G ) ) )
58	32, 57	eqelssd 3198	. . . . 5 |- ( ph -> dom ( S _D ( F oF + G ) ) = X )
59	58	feq2d 5391	. . . 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 2194	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( S _D F ) ` x ) )
62		eqidd 2194	. . 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 14852	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D ( F oF + G ) ) ` x ) = ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
65	64	eqcomd 2199	. . 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 6144	. 2 |- ( ph -> ( ( S _D F ) oF + ( S _D G ) ) = ( S _D ( F oF + G ) ) )
67	66	eqcomd 2199	1 |- ( ph -> ( S _D ( F oF + G ) ) = ( ( S _D F ) oF + ( S _D G ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   {cpr 3619   class class class wbr 4029   dom cdm 4659   o. ccom 4663   Rel wrel 4664   Fun wfun 5248   -->wf 5250   ` cfv 5254  (class class class)co 5918   oFcof 6128   ^pm cpm 6703   CCcc 7870   RRcr 7871   + caddc 7875   - cmin 8190   abscabs 11141   MetOpencmopn 14037   _D cdv 14809

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992  ax-addf 7994

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-map 6704  df-pm 6705  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-xneg 9838  df-xadd 9839  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-rest 12852  df-topgen 12871  df-psmet 14039  df-xmet 14040  df-met 14041  df-bl 14042  df-mopn 14043  df-top 14166  df-topon 14179  df-bases 14211  df-ntr 14264  df-cn 14356  df-cnp 14357  df-tx 14421  df-limced 14810  df-dvap 14811

This theorem is referenced by:  dvmptaddx  14866