Theorem dvimulf 15365

Description: The product 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
dvimulf	|- ( ph -> ( S _D ( F oF x. G ) ) = ( ( ( S _D F ) oF x. G ) oF + ( ( S _D G ) oF x. F ) ) )

Proof of Theorem dvimulf

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

Step	Hyp	Ref	Expression
1		dvaddf.f	. . . . 5 |- ( ph -> F : X --> CC )
2	1	adantr 276	. . . 4 |- ( ( ph /\ x e. X ) -> F : X --> CC )
3		dviaddf.x	. . . . 5 |- ( ph -> X C_ S )
4	3	adantr 276	. . . 4 |- ( ( ph /\ x e. X ) -> X C_ S )
5		dvaddf.g	. . . . 5 |- ( ph -> G : X --> CC )
6	5	adantr 276	. . . 4 |- ( ( ph /\ x e. X ) -> G : X --> CC )
7		dvaddf.s	. . . . 5 |- ( ph -> S e. { RR , CC } )
8	7	adantr 276	. . . 4 |- ( ( ph /\ x e. X ) -> S e. { RR , CC } )
9		dvaddf.df	. . . . . 6 |- ( ph -> dom ( S _D F ) = X )
10	9	eleq2d 2299	. . . . 5 |- ( ph -> ( x e. dom ( S _D F ) <-> x e. X ) )
11	10	biimpar 297	. . . 4 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D F ) )
12		dvaddf.dg	. . . . . 6 |- ( ph -> dom ( S _D G ) = X )
13	12	eleq2d 2299	. . . . 5 |- ( ph -> ( x e. dom ( S _D G ) <-> x e. X ) )
14	13	biimpar 297	. . . 4 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D G ) )
15	2, 4, 6, 8, 11, 14	dvmulxx 15363	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D ( F oF x. G ) ) ` x ) = ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) )
16	15	mpteq2dva 4173	. 2 |- ( ph -> ( x e. X |-> ( ( S _D ( F oF x. G ) ) ` x ) ) = ( x e. X |-> ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) ) )
17		cnex 8111	. . . . . . 7 |- CC e. _V
18	17	a1i 9	. . . . . 6 |- ( ph -> CC e. _V )
19		mulcl 8114	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
20	19	adantl 277	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
21	7, 3	ssexd 4223	. . . . . . 7 |- ( ph -> X e. _V )
22		inidm 3413	. . . . . . 7 |- ( X i^i X ) = X
23	20, 1, 5, 21, 21, 22	off 6221	. . . . . 6 |- ( ph -> ( F oF x. G ) : X --> CC )
24		elpm2r 6803	. . . . . 6 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( ( F oF x. G ) : X --> CC /\ X C_ S ) ) -> ( F oF x. G ) e. ( CC ^pm S ) )
25	18, 7, 23, 3, 24	syl22anc 1272	. . . . 5 |- ( ph -> ( F oF x. G ) e. ( CC ^pm S ) )
26		dvfgg 15347	. . . . 5 |- ( ( S e. { RR , CC } /\ ( F oF x. G ) e. ( CC ^pm S ) ) -> ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC )
27	7, 25, 26	syl2anc 411	. . . 4 |- ( ph -> ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC )
28		recnprss 15346	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
29	7, 28	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
30	29, 23, 3	dvbss 15344	. . . . . 6 |- ( ph -> dom ( S _D ( F oF x. G ) ) C_ X )
31		reldvg 15338	. . . . . . . . 9 |- ( ( S C_ CC /\ ( F oF x. G ) e. ( CC ^pm S ) ) -> Rel ( S _D ( F oF x. G ) ) )
32	29, 25, 31	syl2anc 411	. . . . . . . 8 |- ( ph -> Rel ( S _D ( F oF x. G ) ) )
33	32	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. X ) -> Rel ( S _D ( F oF x. G ) ) )
34	29	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> S C_ CC )
35		elpm2r 6803	. . . . . . . . . . . . 13 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : X --> CC /\ X C_ S ) ) -> F e. ( CC ^pm S ) )
36	18, 7, 1, 3, 35	syl22anc 1272	. . . . . . . . . . . 12 |- ( ph -> F e. ( CC ^pm S ) )
37		dvfgg 15347	. . . . . . . . . . . 12 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
38	7, 36, 37	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
39		ffun 5472	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
40		funfvbrb 5741	. . . . . . . . . . 11 |- ( Fun ( S _D F ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
41	38, 39, 40	3syl 17	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
42	41	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
43	11, 42	mpbid 147	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D F ) ( ( S _D F ) ` x ) )
44		elpm2r 6803	. . . . . . . . . . . . 13 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( G : X --> CC /\ X C_ S ) ) -> G e. ( CC ^pm S ) )
45	18, 7, 5, 3, 44	syl22anc 1272	. . . . . . . . . . . 12 |- ( ph -> G e. ( CC ^pm S ) )
46		dvfgg 15347	. . . . . . . . . . . 12 |- ( ( S e. { RR , CC } /\ G e. ( CC ^pm S ) ) -> ( S _D G ) : dom ( S _D G ) --> CC )
47	7, 45, 46	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
48		ffun 5472	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
49		funfvbrb 5741	. . . . . . . . . . 11 |- ( Fun ( S _D G ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
50	47, 48, 49	3syl 17	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
51	50	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
52	14, 51	mpbid 147	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D G ) ( ( S _D G ) ` x ) )
53		eqid 2229	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
54	2, 4, 6, 34, 43, 52, 53	dvmulxxbr 15361	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D ( F oF x. G ) ) ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) )
55		releldm 4955	. . . . . . 7 |- ( ( Rel ( S _D ( F oF x. G ) ) /\ x ( S _D ( F oF x. G ) ) ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) ) -> x e. dom ( S _D ( F oF x. G ) ) )
56	33, 54, 55	syl2anc 411	. . . . . 6 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D ( F oF x. G ) ) )
57	30, 56	eqelssd 3243	. . . . 5 |- ( ph -> dom ( S _D ( F oF x. G ) ) = X )
58	57	feq2d 5457	. . . 4 |- ( ph -> ( ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC <-> ( S _D ( F oF x. G ) ) : X --> CC ) )
59	27, 58	mpbid 147	. . 3 |- ( ph -> ( S _D ( F oF x. G ) ) : X --> CC )
60	59	feqmptd 5680	. 2 |- ( ph -> ( S _D ( F oF x. G ) ) = ( x e. X |-> ( ( S _D ( F oF x. G ) ) ` x ) ) )
61	9	feq2d 5457	. . . . . 6 |- ( ph -> ( ( S _D F ) : dom ( S _D F ) --> CC <-> ( S _D F ) : X --> CC ) )
62	38, 61	mpbid 147	. . . . 5 |- ( ph -> ( S _D F ) : X --> CC )
63	62	ffvelcdmda 5763	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) e. CC )
64	5	ffvelcdmda 5763	. . . 4 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. CC )
65	63, 64	mulcld 8155	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) e. CC )
66	12	feq2d 5457	. . . . . 6 |- ( ph -> ( ( S _D G ) : dom ( S _D G ) --> CC <-> ( S _D G ) : X --> CC ) )
67	47, 66	mpbid 147	. . . . 5 |- ( ph -> ( S _D G ) : X --> CC )
68	67	ffvelcdmda 5763	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D G ) ` x ) e. CC )
69	1	ffvelcdmda 5763	. . . 4 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. CC )
70	68, 69	mulcld 8155	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D G ) ` x ) x. ( F ` x ) ) e. CC )
71	62	feqmptd 5680	. . . 4 |- ( ph -> ( S _D F ) = ( x e. X |-> ( ( S _D F ) ` x ) ) )
72	5	feqmptd 5680	. . . 4 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
73	21, 63, 64, 71, 72	offval2 6224	. . 3 |- ( ph -> ( ( S _D F ) oF x. G ) = ( x e. X |-> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) ) )
74	67	feqmptd 5680	. . . 4 |- ( ph -> ( S _D G ) = ( x e. X |-> ( ( S _D G ) ` x ) ) )
75	1	feqmptd 5680	. . . 4 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
76	21, 68, 69, 74, 75	offval2 6224	. . 3 |- ( ph -> ( ( S _D G ) oF x. F ) = ( x e. X |-> ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) )
77	21, 65, 70, 73, 76	offval2 6224	. 2 |- ( ph -> ( ( ( S _D F ) oF x. G ) oF + ( ( S _D G ) oF x. F ) ) = ( x e. X |-> ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) ) )
78	16, 60, 77	3eqtr4d 2272	1 |- ( ph -> ( S _D ( F oF x. G ) ) = ( ( ( S _D F ) oF x. G ) oF + ( ( S _D G ) oF x. F ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   {cpr 3667   class class class wbr 4082   |-> cmpt 4144   dom cdm 4716   o. ccom 4720   Rel wrel 4721   Fun wfun 5308   -->wf 5310   ` cfv 5314  (class class class)co 5994   oFcof 6206   ^pm cpm 6786   CCcc 7985   RRcr 7986   + caddc 7990   x. cmul 7992   - cmin 8305   abscabs 11494   MetOpencmopn 14490   _D cdv 15314

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107  ax-addf 8109  ax-mulf 8110

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-of 6208  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-map 6787  df-pm 6788  df-sup 7139  df-inf 7140  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-xneg 9956  df-xadd 9957  df-seqfrec 10657  df-exp 10748  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-rest 13260  df-topgen 13279  df-psmet 14492  df-xmet 14493  df-met 14494  df-bl 14495  df-mopn 14496  df-top 14657  df-topon 14670  df-bases 14702  df-ntr 14755  df-cn 14847  df-cnp 14848  df-tx 14912  df-cncf 15230  df-limced 15315  df-dvap 15316

This theorem is referenced by:  dvexp  15370  dvmptmulx  15379