Theorem dviaddf 15428

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 8156	. . . 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 8155	. . . . . . 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 6834	. . . . . 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 1274	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
10		dvfgg 15411	. . . . 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 5470	. . . 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 6834	. . . . . 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 1274	. . . . 5 |- ( ph -> G e. ( CC ^pm S ) )
18		dvfgg 15411	. . . . 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 5470	. . . 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 4229	. . 3 |- ( ph -> X e. _V )
24		inidm 3416	. . 3 |- ( X i^i X ) = X
25	2, 6, 15, 23, 23, 24	off 6247	. . . . . 6 |- ( ph -> ( F oF + G ) : X --> CC )
26		elpm2r 6834	. . . . . 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 1274	. . . . 5 |- ( ph -> ( F oF + G ) e. ( CC ^pm S ) )
28		dvfgg 15411	. . . . 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 15410	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
31	3, 30	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
32	31, 25, 7	dvbss 15408	. . . . . 6 |- ( ph -> dom ( S _D ( F oF + G ) ) C_ X )
33		reldvg 15402	. . . . . . . . 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 2301	. . . . . . . . . 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 5485	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
43		funfvbrb 5760	. . . . . . . . . . 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 2301	. . . . . . . . . 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 5485	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
50		funfvbrb 5760	. . . . . . . . . . 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 2231	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
55	36, 37, 38, 39, 46, 53, 54	dvaddxxbr 15424	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D ( F oF + G ) ) ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
56		releldm 4967	. . . . . . 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 3246	. . . . 5 |- ( ph -> dom ( S _D ( F oF + G ) ) = X )
59	58	feq2d 5470	. . . 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 2232	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( S _D F ) ` x ) )
62		eqidd 2232	. . 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 15426	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D ( F oF + G ) ) ` x ) = ( ( ( S _D F ) ` x ) + ( ( S _D G ) ` x ) ) )
65	64	eqcomd 2237	. . 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 6248	. 2 |- ( ph -> ( ( S _D F ) oF + ( S _D G ) ) = ( S _D ( F oF + G ) ) )
67	66	eqcomd 2237	1 |- ( ph -> ( S _D ( F oF + G ) ) = ( ( S _D F ) oF + ( S _D G ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   {cpr 3670   class class class wbr 4088   dom cdm 4725   o. ccom 4729   Rel wrel 4730   Fun wfun 5320   -->wf 5322   ` cfv 5326  (class class class)co 6017   oFcof 6232   ^pm cpm 6817   CCcc 8029   RRcr 8030   + caddc 8034   - cmin 8349   abscabs 11557   MetOpencmopn 14554   _D cdv 15378

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151  ax-addf 8153

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-of 6234  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-map 6818  df-pm 6819  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-xneg 10006  df-xadd 10007  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-rest 13323  df-topgen 13342  df-psmet 14556  df-xmet 14557  df-met 14558  df-bl 14559  df-mopn 14560  df-top 14721  df-topon 14734  df-bases 14766  df-ntr 14819  df-cn 14911  df-cnp 14912  df-tx 14976  df-limced 15379  df-dvap 15380

This theorem is referenced by:  dvmptaddx  15442