Theorem dvimulf 14942

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 2266	. . . . 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 2266	. . . . 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 14940	. . 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 4123	. 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 8003	. . . . . . 7 |- CC e. _V
18	17	a1i 9	. . . . . 6 |- ( ph -> CC e. _V )
19		mulcl 8006	. . . . . . . 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 4173	. . . . . . 7 |- ( ph -> X e. _V )
22		inidm 3372	. . . . . . 7 |- ( X i^i X ) = X
23	20, 1, 5, 21, 21, 22	off 6148	. . . . . 6 |- ( ph -> ( F oF x. G ) : X --> CC )
24		elpm2r 6725	. . . . . 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 14924	. . . . 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 14923	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
29	7, 28	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
30	29, 23, 3	dvbss 14921	. . . . . 6 |- ( ph -> dom ( S _D ( F oF x. G ) ) C_ X )
31		reldvg 14915	. . . . . . . . 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 6725	. . . . . . . . . . . . 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 14924	. . . . . . . . . . . 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 5410	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
40		funfvbrb 5675	. . . . . . . . . . 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 6725	. . . . . . . . . . . . 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 14924	. . . . . . . . . . . 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 5410	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
49		funfvbrb 5675	. . . . . . . . . . 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 2196	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
54	2, 4, 6, 34, 43, 52, 53	dvmulxxbr 14938	. . . . . . 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 4901	. . . . . . 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 3202	. . . . 5 |- ( ph -> dom ( S _D ( F oF x. G ) ) = X )
58	57	feq2d 5395	. . . 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 5614	. 2 |- ( ph -> ( S _D ( F oF x. G ) ) = ( x e. X |-> ( ( S _D ( F oF x. G ) ) ` x ) ) )
61	9	feq2d 5395	. . . . . 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 5697	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) e. CC )
64	5	ffvelcdmda 5697	. . . 4 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. CC )
65	63, 64	mulcld 8047	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) e. CC )
66	12	feq2d 5395	. . . . . 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 5697	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D G ) ` x ) e. CC )
69	1	ffvelcdmda 5697	. . . 4 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. CC )
70	68, 69	mulcld 8047	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D G ) ` x ) x. ( F ` x ) ) e. CC )
71	62	feqmptd 5614	. . . 4 |- ( ph -> ( S _D F ) = ( x e. X |-> ( ( S _D F ) ` x ) ) )
72	5	feqmptd 5614	. . . 4 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
73	21, 63, 64, 71, 72	offval2 6151	. . 3 |- ( ph -> ( ( S _D F ) oF x. G ) = ( x e. X |-> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) ) )
74	67	feqmptd 5614	. . . 4 |- ( ph -> ( S _D G ) = ( x e. X |-> ( ( S _D G ) ` x ) ) )
75	1	feqmptd 5614	. . . 4 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
76	21, 68, 69, 74, 75	offval2 6151	. . 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 6151	. 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 2239	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 2167   _Vcvv 2763   C_ wss 3157   {cpr 3623   class class class wbr 4033   |-> cmpt 4094   dom cdm 4663   o. ccom 4667   Rel wrel 4668   Fun wfun 5252   -->wf 5254   ` cfv 5258  (class class class)co 5922   oFcof 6133   ^pm cpm 6708   CCcc 7877   RRcr 7878   + caddc 7882   x. cmul 7884   - cmin 8197   abscabs 11162   MetOpencmopn 14097   _D cdv 14891

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999  ax-addf 8001  ax-mulf 8002

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-map 6709  df-pm 6710  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-rest 12912  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102  df-mopn 14103  df-top 14234  df-topon 14247  df-bases 14279  df-ntr 14332  df-cn 14424  df-cnp 14425  df-tx 14489  df-cncf 14807  df-limced 14892  df-dvap 14893

This theorem is referenced by:  dvexp  14947  dvmptmulx  14956