Theorem dviaddf 14208

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 7938	. . . 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 7937	. . . . . . 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 6668	. . . . . 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 1239	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
10		dvfgg 14196	. . . . 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 5355	. . . 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 6668	. . . . . 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 1239	. . . . 5 |- ( ph -> G e. ( CC ^pm S ) )
18		dvfgg 14196	. . . . 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 5355	. . . 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 4145	. . 3 |- ( ph -> X e. _V )
24		inidm 3346	. . 3 |- ( X i^i X ) = X
25	2, 6, 15, 23, 23, 24	off 6097	. . . . . 6 |- ( ph -> ( F oF + G ) : X --> CC )
26		elpm2r 6668	. . . . . 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 1239	. . . . 5 |- ( ph -> ( F oF + G ) e. ( CC ^pm S ) )
28		dvfgg 14196	. . . . 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 14195	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
31	3, 30	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
32	31, 25, 7	dvbss 14193	. . . . . 6 |- ( ph -> dom ( S _D ( F oF + G ) ) C_ X )
33		reldvg 14187	. . . . . . . . 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 2247	. . . . . . . . . 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 5370	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
43		funfvbrb 5631	. . . . . . . . . . 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 2247	. . . . . . . . . 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 5370	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
50		funfvbrb 5631	. . . . . . . . . . 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 2177	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
55	36, 37, 38, 39, 46, 53, 54	dvaddxxbr 14204	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D ( F oF + G ) ) ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
56		releldm 4864	. . . . . . 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 3176	. . . . 5 |- ( ph -> dom ( S _D ( F oF + G ) ) = X )
59	58	feq2d 5355	. . . 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 2178	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( S _D F ) ` x ) )
62		eqidd 2178	. . 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 14206	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D ( F oF + G ) ) ` x ) = ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
65	64	eqcomd 2183	. . 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 6098	. 2 |- ( ph -> ( ( S _D F ) oF + ( S _D G ) ) = ( S _D ( F oF + G ) ) )
67	66	eqcomd 2183	1 |- ( ph -> ( S _D ( F oF + G ) ) = ( ( S _D F ) oF + ( S _D G ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2739   C_ wss 3131   {cpr 3595   class class class wbr 4005   dom cdm 4628   o. ccom 4632   Rel wrel 4633   Fun wfun 5212   -->wf 5214   ` cfv 5218  (class class class)co 5877   oFcof 6083   ^pm cpm 6651   CCcc 7811   RRcr 7812   + caddc 7816   - cmin 8130   abscabs 11008   MetOpencmopn 13484   _D cdv 14163

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933  ax-addf 7935

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-of 6085  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-map 6652  df-pm 6653  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-xneg 9774  df-xadd 9775  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-rest 12695  df-topgen 12714  df-psmet 13486  df-xmet 13487  df-met 13488  df-bl 13489  df-mopn 13490  df-top 13537  df-topon 13550  df-bases 13582  df-ntr 13635  df-cn 13727  df-cnp 13728  df-tx 13792  df-limced 14164  df-dvap 14165

This theorem is referenced by:  dvmptaddx  14220