Theorem dvimulf 14655

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 2259	. . . . 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 2259	. . . . 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 14653	. . 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 4111	. 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 7970	. . . . . . 7 |- CC e. _V
18	17	a1i 9	. . . . . 6 |- ( ph -> CC e. _V )
19		mulcl 7973	. . . . . . . 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 4161	. . . . . . 7 |- ( ph -> X e. _V )
22		inidm 3359	. . . . . . 7 |- ( X i^i X ) = X
23	20, 1, 5, 21, 21, 22	off 6123	. . . . . 6 |- ( ph -> ( F oF x. G ) : X --> CC )
24		elpm2r 6696	. . . . . 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 1250	. . . . 5 |- ( ph -> ( F oF x. G ) e. ( CC ^pm S ) )
26		dvfgg 14642	. . . . 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 14641	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
29	7, 28	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
30	29, 23, 3	dvbss 14639	. . . . . 6 |- ( ph -> dom ( S _D ( F oF x. G ) ) C_ X )
31		reldvg 14633	. . . . . . . . 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 6696	. . . . . . . . . . . . 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 1250	. . . . . . . . . . . 12 |- ( ph -> F e. ( CC ^pm S ) )
37		dvfgg 14642	. . . . . . . . . . . 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 5390	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
40		funfvbrb 5653	. . . . . . . . . . 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 6696	. . . . . . . . . . . . 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 1250	. . . . . . . . . . . 12 |- ( ph -> G e. ( CC ^pm S ) )
46		dvfgg 14642	. . . . . . . . . . . 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 5390	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
49		funfvbrb 5653	. . . . . . . . . . 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 2189	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
54	2, 4, 6, 34, 43, 52, 53	dvmulxxbr 14651	. . . . . . 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 4883	. . . . . . 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 3189	. . . . 5 |- ( ph -> dom ( S _D ( F oF x. G ) ) = X )
58	57	feq2d 5375	. . . 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 5593	. 2 |- ( ph -> ( S _D ( F oF x. G ) ) = ( x e. X |-> ( ( S _D ( F oF x. G ) ) ` x ) ) )
61	9	feq2d 5375	. . . . . 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 5675	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) e. CC )
64	5	ffvelcdmda 5675	. . . 4 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. CC )
65	63, 64	mulcld 8013	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) e. CC )
66	12	feq2d 5375	. . . . . 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 5675	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D G ) ` x ) e. CC )
69	1	ffvelcdmda 5675	. . . 4 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. CC )
70	68, 69	mulcld 8013	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D G ) ` x ) x. ( F ` x ) ) e. CC )
71	62	feqmptd 5593	. . . 4 |- ( ph -> ( S _D F ) = ( x e. X |-> ( ( S _D F ) ` x ) ) )
72	5	feqmptd 5593	. . . 4 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
73	21, 63, 64, 71, 72	offval2 6126	. . 3 |- ( ph -> ( ( S _D F ) oF x. G ) = ( x e. X |-> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) ) )
74	67	feqmptd 5593	. . . 4 |- ( ph -> ( S _D G ) = ( x e. X |-> ( ( S _D G ) ` x ) ) )
75	1	feqmptd 5593	. . . 4 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
76	21, 68, 69, 74, 75	offval2 6126	. . 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 6126	. 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 2232	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 1364   e. wcel 2160   _Vcvv 2752   C_ wss 3144   {cpr 3611   class class class wbr 4021   |-> cmpt 4082   dom cdm 4647   o. ccom 4651   Rel wrel 4652   Fun wfun 5232   -->wf 5234   ` cfv 5238  (class class class)co 5900   oFcof 6108   ^pm cpm 6679   CCcc 7844   RRcr 7845   + caddc 7849   x. cmul 7851   - cmin 8163   abscabs 11047   MetOpencmopn 13879   _D cdv 14609

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964  ax-arch 7965  ax-caucvg 7966  ax-addf 7968  ax-mulf 7969

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-isom 5247  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-of 6110  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-map 6680  df-pm 6681  df-sup 7017  df-inf 7018  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-n0 9212  df-z 9289  df-uz 9564  df-q 9656  df-rp 9690  df-xneg 9808  df-xadd 9809  df-seqfrec 10485  df-exp 10560  df-cj 10892  df-re 10893  df-im 10894  df-rsqrt 11048  df-abs 11049  df-rest 12757  df-topgen 12776  df-psmet 13881  df-xmet 13882  df-met 13883  df-bl 13884  df-mopn 13885  df-top 13983  df-topon 13996  df-bases 14028  df-ntr 14081  df-cn 14173  df-cnp 14174  df-tx 14238  df-cncf 14543  df-limced 14610  df-dvap 14611

This theorem is referenced by:  dvexp  14660  dvmptmulx  14667