Theorem dvimulf 13310

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 274	. . . 4 |- ( ( ph /\ x e. X ) -> F : X --> CC )
3		dviaddf.x	. . . . 5 |- ( ph -> X C_ S )
4	3	adantr 274	. . . 4 |- ( ( ph /\ x e. X ) -> X C_ S )
5		dvaddf.g	. . . . 5 |- ( ph -> G : X --> CC )
6	5	adantr 274	. . . 4 |- ( ( ph /\ x e. X ) -> G : X --> CC )
7		dvaddf.s	. . . . 5 |- ( ph -> S e. { RR , CC } )
8	7	adantr 274	. . . 4 |- ( ( ph /\ x e. X ) -> S e. { RR , CC } )
9		dvaddf.df	. . . . . 6 |- ( ph -> dom ( S _D F ) = X )
10	9	eleq2d 2236	. . . . 5 |- ( ph -> ( x e. dom ( S _D F ) <-> x e. X ) )
11	10	biimpar 295	. . . 4 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D F ) )
12		dvaddf.dg	. . . . . 6 |- ( ph -> dom ( S _D G ) = X )
13	12	eleq2d 2236	. . . . 5 |- ( ph -> ( x e. dom ( S _D G ) <-> x e. X ) )
14	13	biimpar 295	. . . 4 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D G ) )
15	2, 4, 6, 8, 11, 14	dvmulxx 13308	. . 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 4072	. 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 7877	. . . . . . 7 |- CC e. _V
18	17	a1i 9	. . . . . 6 |- ( ph -> CC e. _V )
19		mulcl 7880	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
20	19	adantl 275	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
21	7, 3	ssexd 4122	. . . . . . 7 |- ( ph -> X e. _V )
22		inidm 3331	. . . . . . 7 |- ( X i^i X ) = X
23	20, 1, 5, 21, 21, 22	off 6062	. . . . . 6 |- ( ph -> ( F oF x. G ) : X --> CC )
24		elpm2r 6632	. . . . . 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 1229	. . . . 5 |- ( ph -> ( F oF x. G ) e. ( CC ^pm S ) )
26		dvfgg 13297	. . . . 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 409	. . . 4 |- ( ph -> ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC )
28		recnprss 13296	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
29	7, 28	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
30	29, 23, 3	dvbss 13294	. . . . . 6 |- ( ph -> dom ( S _D ( F oF x. G ) ) C_ X )
31		reldvg 13288	. . . . . . . . 9 |- ( ( S C_ CC /\ ( F oF x. G ) e. ( CC ^pm S ) ) -> Rel ( S _D ( F oF x. G ) ) )
32	29, 25, 31	syl2anc 409	. . . . . . . 8 |- ( ph -> Rel ( S _D ( F oF x. G ) ) )
33	32	adantr 274	. . . . . . 7 |- ( ( ph /\ x e. X ) -> Rel ( S _D ( F oF x. G ) ) )
34	29	adantr 274	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> S C_ CC )
35		elpm2r 6632	. . . . . . . . . . . . 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 1229	. . . . . . . . . . . 12 |- ( ph -> F e. ( CC ^pm S ) )
37		dvfgg 13297	. . . . . . . . . . . 12 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
38	7, 36, 37	syl2anc 409	. . . . . . . . . . 11 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
39		ffun 5340	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
40		funfvbrb 5598	. . . . . . . . . . 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 274	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
43	11, 42	mpbid 146	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D F ) ( ( S _D F ) ` x ) )
44		elpm2r 6632	. . . . . . . . . . . . 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 1229	. . . . . . . . . . . 12 |- ( ph -> G e. ( CC ^pm S ) )
46		dvfgg 13297	. . . . . . . . . . . 12 |- ( ( S e. { RR , CC } /\ G e. ( CC ^pm S ) ) -> ( S _D G ) : dom ( S _D G ) --> CC )
47	7, 45, 46	syl2anc 409	. . . . . . . . . . 11 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
48		ffun 5340	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
49		funfvbrb 5598	. . . . . . . . . . 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 274	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
52	14, 51	mpbid 146	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D G ) ( ( S _D G ) ` x ) )
53		eqid 2165	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
54	2, 4, 6, 34, 43, 52, 53	dvmulxxbr 13306	. . . . . . 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 4839	. . . . . . 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 409	. . . . . 6 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D ( F oF x. G ) ) )
57	30, 56	eqelssd 3161	. . . . 5 |- ( ph -> dom ( S _D ( F oF x. G ) ) = X )
58	57	feq2d 5325	. . . 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 146	. . 3 |- ( ph -> ( S _D ( F oF x. G ) ) : X --> CC )
60	59	feqmptd 5539	. 2 |- ( ph -> ( S _D ( F oF x. G ) ) = ( x e. X |-> ( ( S _D ( F oF x. G ) ) ` x ) ) )
61	9	feq2d 5325	. . . . . 6 |- ( ph -> ( ( S _D F ) : dom ( S _D F ) --> CC <-> ( S _D F ) : X --> CC ) )
62	38, 61	mpbid 146	. . . . 5 |- ( ph -> ( S _D F ) : X --> CC )
63	62	ffvelrnda 5620	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) e. CC )
64	5	ffvelrnda 5620	. . . 4 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. CC )
65	63, 64	mulcld 7919	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) e. CC )
66	12	feq2d 5325	. . . . . 6 |- ( ph -> ( ( S _D G ) : dom ( S _D G ) --> CC <-> ( S _D G ) : X --> CC ) )
67	47, 66	mpbid 146	. . . . 5 |- ( ph -> ( S _D G ) : X --> CC )
68	67	ffvelrnda 5620	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D G ) ` x ) e. CC )
69	1	ffvelrnda 5620	. . . 4 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. CC )
70	68, 69	mulcld 7919	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D G ) ` x ) x. ( F ` x ) ) e. CC )
71	62	feqmptd 5539	. . . 4 |- ( ph -> ( S _D F ) = ( x e. X |-> ( ( S _D F ) ` x ) ) )
72	5	feqmptd 5539	. . . 4 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
73	21, 63, 64, 71, 72	offval2 6065	. . 3 |- ( ph -> ( ( S _D F ) oF x. G ) = ( x e. X |-> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) ) )
74	67	feqmptd 5539	. . . 4 |- ( ph -> ( S _D G ) = ( x e. X |-> ( ( S _D G ) ` x ) ) )
75	1	feqmptd 5539	. . . 4 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
76	21, 68, 69, 74, 75	offval2 6065	. . 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 6065	. 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 2208	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 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   {cpr 3577   class class class wbr 3982   |-> cmpt 4043   dom cdm 4604   o. ccom 4608   Rel wrel 4609   Fun wfun 5182   -->wf 5184   ` cfv 5188  (class class class)co 5842   oFcof 6048   ^pm cpm 6615   CCcc 7751   RRcr 7752   + caddc 7756   x. cmul 7758   - cmin 8069   abscabs 10939   MetOpencmopn 12625   _D cdv 13264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-addf 7875  ax-mulf 7876

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-map 6616  df-pm 6617  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266

This theorem is referenced by:  dvexp  13315  dvmptmulx  13322