Theorem muldisc 14824

Description: Multiplication by a difference of scalars.

Hypotheses

Ref	Expression
muldisc.1	|- X = ran (1st` (1st` R))
muldisc.2	|- +t = (1st` (1st` R))
muldisc.4	|- +w = (1st` (2nd` R))
muldisc.5	|- .w = (2nd` (2nd` R))
muldisc.6	|- W = ran (1st` (2nd` R))
muldisc.7	|- -t = ( /g ` +t )
muldisc.8	|- -w = ( /g ` +w )
muldisc.9	|- .t = (2nd` (1st` R))

Assertion

Ref	Expression
muldisc	|- ((R e. Vec /\ <.+t , .t >. e. Ring) -> A.u e. W A.x e. X A.y e. X ((x-t y).w u) = ((x.w u)-w (y.w u)))

Distinct variable groups:   u,+_t ,x,y   u,._t ,x,y   u,R,x,y   x,W,y   y,X

Proof of Theorem muldisc

Step	Hyp	Ref	Expression
1		muldisc.2	. . . . . . . . . . . . . 14 |- +t = (1st` (1st` R))
2		fvex 4689	. . . . . . . . . . . . . 14 |- (1st` (1st` R)) e. _V
3	1, 2	eqeltri 1967	. . . . . . . . . . . . 13 |- +t e. _V
4	3	op1st 5026	. . . . . . . . . . . 12 |- (1st` <.+t , .t >.) = +t
5	4	eqcomi 1888	. . . . . . . . . . 11 |- +t = (1st` <.+t , .t >.)
6	5	ringgrp 9476	. . . . . . . . . 10 |- (<.+t , .t >. e. Ring -> +t e. Grp)
7	6	adantl 424	. . . . . . . . 9 |- ((R e. Vec /\ <.+t , .t >. e. Ring) -> +t e. Grp)
8	7	ad2antrr 440	. . . . . . . 8 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> +t e. Grp)
9		muldisc.1	. . . . . . . . . . . 12 |- X = ran (1st` (1st` R))
10	9	eleq2i 1961	. . . . . . . . . . 11 |- (x e. X <-> x e. ran (1st` (1st` R)))
11	10	biimpi 168	. . . . . . . . . 10 |- (x e. X -> x e. ran (1st` (1st` R)))
12	11	adantl 424	. . . . . . . . 9 |- ((u e. W /\ x e. X) -> x e. ran (1st` (1st` R)))
13	12	ad2antlr 441	. . . . . . . 8 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> x e. ran (1st` (1st` R)))
14	9	eleq2i 1961	. . . . . . . . . 10 |- (y e. X <-> y e. ran (1st` (1st` R)))
15	14	biimpi 168	. . . . . . . . 9 |- (y e. X -> y e. ran (1st` (1st` R)))
16	15	adantl 424	. . . . . . . 8 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> y e. ran (1st` (1st` R)))
17	1	rneqi 4187	. . . . . . . . . 10 |- ran +t = ran (1st` (1st` R))
18	17	eqcomi 1888	. . . . . . . . 9 |- ran (1st` (1st` R)) = ran +t
19		eqid 1884	. . . . . . . . 9 |- (inv` +t ) = (inv` +t )
20		muldisc.7	. . . . . . . . 9 |- -t = ( /g ` +t )
21	18, 19, 20	grpdivval 9367	. . . . . . . 8 |- ((+t e. Grp /\ x e. ran (1st` (1st` R)) /\ y e. ran (1st` (1st` R))) -> (x-t y) = (x+t ((inv` +t )` y)))
22	8, 13, 16, 21	syl111anc 1100	. . . . . . 7 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> (x-t y) = (x+t ((inv` +t )` y)))
23	22	opreq1d 4897	. . . . . 6 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> ((x-t y).w u) = ((x+t ((inv` +t )` y)).w u))
24		simplll 452	. . . . . . 7 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> R e. Vec)
25		simplrl 454	. . . . . . 7 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> u e. W)
26		simplrr 455	. . . . . . 7 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> x e. X)
27	1	fveq2i 4684	. . . . . . . . 9 |- (inv` +t ) = (inv` (1st` (1st` R)))
28	9, 27	grpinvcl 9352	. . . . . . . 8 |- (((1st` (1st` R)) e. Grp /\ y e. X) -> ((inv` +t )` y) e. X)
29	6, 1	syl5eqelr 1976	. . . . . . . . 9 |- (<.+t , .t >. e. Ring -> (1st` (1st` R)) e. Grp)
30	29	ad2antlr 441	. . . . . . . 8 |- (((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) -> (1st` (1st` R)) e. Grp)
31	28, 30	sylan 497	. . . . . . 7 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> ((inv` +t )` y) e. X)
32		muldisc.4	. . . . . . . 8 |- +w = (1st` (2nd` R))
33		muldisc.5	. . . . . . . 8 |- .w = (2nd` (2nd` R))
34		muldisc.6	. . . . . . . 8 |- W = ran (1st` (2nd` R))
35	9, 1, 32, 33, 34	vecax5b 14802	. . . . . . 7 |- ((R e. Vec /\ (u e. W /\ x e. X /\ ((inv` +t )` y) e. X)) -> ((x+t ((inv` +t )` y)).w u) = ((x.w u)+w (((inv` +t )` y).w u)))
36	24, 25, 26, 31, 35	syl13anc 1102	. . . . . 6 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> ((x+t ((inv` +t )` y)).w u) = ((x.w u)+w (((inv` +t )` y).w u)))
37		simpllr 453	. . . . . . . 8 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> <.+t , .t >. e. Ring)
38	1	eqcomi 1888	. . . . . . . . . 10 |- (1st` (1st` R)) = +t
39	38	rneqi 4187	. . . . . . . . 9 |- ran (1st` (1st` R)) = ran +t
40	32	fveq2i 4684	. . . . . . . . 9 |- (inv` +w ) = (inv` (1st` (2nd` R)))
41		muldisc.9	. . . . . . . . 9 |- .t = (2nd` (1st` R))
42	39, 34, 33, 19, 40, 1, 41	mulinvsca 14823	. . . . . . . 8 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (y e. ran (1st` (1st` R)) /\ u e. W)) -> (((inv` +t )` y).w u) = ((inv` +w )` (y.w u)))
43	24, 37, 16, 25, 42	syl112anc 1104	. . . . . . 7 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> (((inv` +t )` y).w u) = ((inv` +w )` (y.w u)))
44	43	opreq2d 4898	. . . . . 6 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> ((x.w u)+w (((inv` +t )` y).w u)) = ((x.w u)+w ((inv` +w )` (y.w u))))
45	23, 36, 44	3eqtrd 1929	. . . . 5 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> ((x-t y).w u) = ((x.w u)+w ((inv` +w )` (y.w u))))
46	32	vecax1 14796	. . . . . . . . 9 |- (R e. Vec -> +w e. Abel)
47		ablgrp 9410	. . . . . . . . 9 |- (+w e. Abel -> +w e. Grp)
48	46, 47	syl 12	. . . . . . . 8 |- (R e. Vec -> +w e. Grp)
49	48	adantr 425	. . . . . . 7 |- ((R e. Vec /\ <.+t , .t >. e. Ring) -> +w e. Grp)
50	49	ad2antrr 440	. . . . . 6 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> +w e. Grp)
51	9, 34, 33	prvs 14821	. . . . . . . . . . . . 13 |- ((R e. Vec /\ x e. X /\ u e. W) -> (x.w u) e. W)
52	51	3exp 1066	. . . . . . . . . . . 12 |- (R e. Vec -> (x e. X -> (u e. W -> (x.w u) e. W)))
53	52	com3l 38	. . . . . . . . . . 11 |- (x e. X -> (u e. W -> (R e. Vec -> (x.w u) e. W)))
54	53	impcom 378	. . . . . . . . . 10 |- ((u e. W /\ x e. X) -> (R e. Vec -> (x.w u) e. W))
55	54	com12 14	. . . . . . . . 9 |- (R e. Vec -> ((u e. W /\ x e. X) -> (x.w u) e. W))
56	55	adantr 425	. . . . . . . 8 |- ((R e. Vec /\ <.+t , .t >. e. Ring) -> ((u e. W /\ x e. X) -> (x.w u) e. W))
57	56	imp 377	. . . . . . 7 |- (((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) -> (x.w u) e. W)
58	57	adantr 425	. . . . . 6 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> (x.w u) e. W)
59	9, 34, 33	prvs 14821	. . . . . . . . . . . 12 |- ((R e. Vec /\ y e. X /\ u e. W) -> (y.w u) e. W)
60	59	3exp 1066	. . . . . . . . . . 11 |- (R e. Vec -> (y e. X -> (u e. W -> (y.w u) e. W)))
61	60	adantr 425	. . . . . . . . . 10 |- ((R e. Vec /\ <.+t , .t >. e. Ring) -> (y e. X -> (u e. W -> (y.w u) e. W)))
62	61	com3r 39	. . . . . . . . 9 |- (u e. W -> ((R e. Vec /\ <.+t , .t >. e. Ring) -> (y e. X -> (y.w u) e. W)))
63	62	adantr 425	. . . . . . . 8 |- ((u e. W /\ x e. X) -> ((R e. Vec /\ <.+t , .t >. e. Ring) -> (y e. X -> (y.w u) e. W)))
64	63	impcom 378	. . . . . . 7 |- (((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) -> (y e. X -> (y.w u) e. W))
65	64	imp 377	. . . . . 6 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> (y.w u) e. W)
66	32	eqcomi 1888	. . . . . . . . 9 |- (1st` (2nd` R)) = +w
67	66	rneqi 4187	. . . . . . . 8 |- ran (1st` (2nd` R)) = ran +w
68	34, 67	eqtri 1908	. . . . . . 7 |- W = ran +w
69		eqid 1884	. . . . . . 7 |- (inv` +w ) = (inv` +w )
70		muldisc.8	. . . . . . 7 |- -w = ( /g ` +w )
71	68, 69, 70	grpdivval 9367	. . . . . 6 |- ((+w e. Grp /\ (x.w u) e. W /\ (y.w u) e. W) -> ((x.w u)-w (y.w u)) = ((x.w u)+w ((inv` +w )` (y.w u))))
72	50, 58, 65, 71	syl111anc 1100	. . . . 5 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> ((x.w u)-w (y.w u)) = ((x.w u)+w ((inv` +w )` (y.w u))))
73	45, 72	eqtr4d 1928	. . . 4 |- ((((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) /\ y e. X) -> ((x-t y).w u) = ((x.w u)-w (y.w u)))
74	73	r19.21aiva 2176	. . 3 |- (((R e. Vec /\ <.+t , .t >. e. Ring) /\ (u e. W /\ x e. X)) -> A.y e. X ((x-t y).w u) = ((x.w u)-w (y.w u)))
75	74	ex 402	. 2 |- ((R e. Vec /\ <.+t , .t >. e. Ring) -> ((u e. W /\ x e. X) -> A.y e. X ((x-t y).w u) = ((x.w u)-w (y.w u))))
76	75	r19.21aivv 2183	1 |- ((R e. Vec /\ <.+t , .t >. e. Ring) -> A.u e. W A.x e. X A.y e. X ((x-t y).w u) = ((x.w u)-w (y.w u)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  <.cop 3046  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  invcgn 9313   /g cgs 9314  Abelcabl 9407  Ringcring 9463  Veccvec 14792

This theorem is referenced by:  vecax5c 14825

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-ring 9464  df-vec 14793

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