Theorem mulinvsca 14823

Description: Multiplication by the inverse of a scalar.

Hypotheses

Ref	Expression
mulinvsca.1	|- X = ran +t
mulinvsca.2	|- W = ran (1st` (2nd` R))
mulinvsca.3	|- .w = (2nd` (2nd` R))
mulinvsca.4	|- ~t = (inv` +t )
mulinvsca.5	|- ~w = (inv` (1st` (2nd` R)))
mulinvsca.6	|- +t = (1st` (1st` R))
mulinvsca.7	|- .t = (2nd` (1st` R))

Assertion

Ref	Expression
mulinvsca	|- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((~t ` A).w U) = (~w ` (A.w U)))

Proof of Theorem mulinvsca

Step	Hyp	Ref	Expression
1		simp1 876	. . . . 5 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> R e. Vec)
2		eqid 1884	. . . . . . . 8 |- (1st` (2nd` R)) = (1st` (2nd` R))
3		mulinvsca.3	. . . . . . . 8 |- .w = (2nd` (2nd` R))
4		mulinvsca.2	. . . . . . . 8 |- W = ran (1st` (2nd` R))
5		mulinvsca.1	. . . . . . . 8 |- X = ran +t
6		mulinvsca.6	. . . . . . . 8 |- +t = (1st` (1st` R))
7	2, 3, 4, 5, 6	prodvs 14811	. . . . . . 7 |- ((R e. Vec /\ A e. X /\ U e. W) -> (A.w U) e. W)
8	7	3expb 1068	. . . . . 6 |- ((R e. Vec /\ (A e. X /\ U e. W)) -> (A.w U) e. W)
9	8	3adant2 895	. . . . 5 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (A.w U) e. W)
10	1, 9	jca 310	. . . 4 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (R e. Vec /\ (A.w U) e. W))
11		eqid 1884	. . . . . 6 |- (Id` (1st` (2nd` R))) = (Id` (1st` (2nd` R)))
12		eqid 1884	. . . . . 6 |- ( /g ` (1st` (2nd` R))) = ( /g ` (1st` (2nd` R)))
13	11, 2, 12, 4	vwit 14814	. . . . 5 |- ((R e. Vec /\ (A.w U) e. W) -> ((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = (Id` (1st` (2nd` R))))
14		simpl 346	. . . . . . 7 |- ((((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = (Id` (1st` (2nd` R))) /\ (R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W))) -> ((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = (Id` (1st` (2nd` R))))
15		eqid 1884	. . . . . . . . . . 11 |- (Id` +t ) = (Id` +t )
16		mulinvsca.7	. . . . . . . . . . 11 |- .t = (2nd` (1st` R))
17	4, 15, 6, 16, 3, 11	mulveczer 14822	. . . . . . . . . 10 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ U e. W) -> ((Id` +t ).w U) = (Id` (1st` (2nd` R))))
18	17	3adant3l 1094	. . . . . . . . 9 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((Id` +t ).w U) = (Id` (1st` (2nd` R))))
19		fvex 4689	. . . . . . . . . . . . . . . . 17 |- (1st` (1st` R)) e. _V
20	6, 19	eqeltri 1967	. . . . . . . . . . . . . . . 16 |- +t e. _V
21	20	op1st 5026	. . . . . . . . . . . . . . 15 |- (1st` <.+t , .t >.) = +t
22	21	eqcomi 1888	. . . . . . . . . . . . . 14 |- +t = (1st` <.+t , .t >.)
23	22	ringgrp 9476	. . . . . . . . . . . . 13 |- (<.+t , .t >. e. Ring -> +t e. Grp)
24	23	3ad2ant2 898	. . . . . . . . . . . 12 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> +t e. Grp)
25		simp3l 904	. . . . . . . . . . . 12 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> A e. X)
26		mulinvsca.4	. . . . . . . . . . . . 13 |- ~t = (inv` +t )
27	5, 15, 26	grprinv 9355	. . . . . . . . . . . 12 |- ((+t e. Grp /\ A e. X) -> (A+t (~t ` A)) = (Id` +t ))
28	24, 25, 27	syl11anc 524	. . . . . . . . . . 11 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (A+t (~t ` A)) = (Id` +t ))
29	28	eqcomd 1889	. . . . . . . . . 10 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (Id` +t ) = (A+t (~t ` A)))
30	29	opreq1d 4897	. . . . . . . . 9 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((Id` +t ).w U) = ((A+t (~t ` A)).w U))
31	18, 30	eqtr3d 1927	. . . . . . . 8 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (Id` (1st` (2nd` R))) = ((A+t (~t ` A)).w U))
32	31	adantl 424	. . . . . . 7 |- ((((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = (Id` (1st` (2nd` R))) /\ (R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W))) -> (Id` (1st` (2nd` R))) = ((A+t (~t ` A)).w U))
33	14, 32	eqtrd 1925	. . . . . 6 |- ((((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = (Id` (1st` (2nd` R))) /\ (R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W))) -> ((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = ((A+t (~t ` A)).w U))
34	33	ex 402	. . . . 5 |- (((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = (Id` (1st` (2nd` R))) -> ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = ((A+t (~t ` A)).w U)))
35	13, 34	syl 12	. . . 4 |- ((R e. Vec /\ (A.w U) e. W) -> ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = ((A+t (~t ` A)).w U)))
36	10, 35	mpcom 60	. . 3 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = ((A+t (~t ` A)).w U))
37		simp3r 905	. . . . 5 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> U e. W)
38	6	rneqi 4187	. . . . . . 7 |- ran +t = ran (1st` (1st` R))
39	5, 38	eqtri 1908	. . . . . 6 |- X = ran (1st` (1st` R))
40	39, 4, 3	prvs 14821	. . . . 5 |- ((R e. Vec /\ A e. X /\ U e. W) -> (A.w U) e. W)
41	1, 25, 37, 40	syl111anc 1100	. . . 4 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (A.w U) e. W)
42		mulinvsca.5	. . . . 5 |- ~w = (inv` (1st` (2nd` R)))
43	11, 2, 12, 4, 42	sub2vec 14815	. . . 4 |- ((R e. Vec /\ ((A.w U) e. W /\ (A.w U) e. W)) -> ((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = ((A.w U)(1st` (2nd` R))(~w ` (A.w U))))
44	1, 41, 41, 43	syl12anc 1098	. . 3 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((A.w U)( /g ` (1st` (2nd` R)))(A.w U)) = ((A.w U)(1st` (2nd` R))(~w ` (A.w U))))
45	5, 26	rnginvcl 14770	. . . . . 6 |- ((<.+t , .t >. e. Ring /\ A e. X) -> (~t ` A) e. X)
46	45	adantrr 431	. . . . 5 |- ((<.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (~t ` A) e. X)
47	46	3adant1 894	. . . 4 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (~t ` A) e. X)
48	39, 6, 2, 3, 4	vecax5b 14802	. . . 4 |- ((R e. Vec /\ (U e. W /\ A e. X /\ (~t ` A) e. X)) -> ((A+t (~t ` A)).w U) = ((A.w U)(1st` (2nd` R))((~t ` A).w U)))
49	1, 37, 25, 47, 48	syl13anc 1102	. . 3 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((A+t (~t ` A)).w U) = ((A.w U)(1st` (2nd` R))((~t ` A).w U)))
50	36, 44, 49	3eqtr3rd 1936	. 2 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((A.w U)(1st` (2nd` R))((~t ` A).w U)) = ((A.w U)(1st` (2nd` R))(~w ` (A.w U))))
51	2	vecax1 14796	. . . . 5 |- (R e. Vec -> (1st` (2nd` R)) e. Abel)
52		ablgrp 9410	. . . . 5 |- ((1st` (2nd` R)) e. Abel -> (1st` (2nd` R)) e. Grp)
53	51, 52	syl 12	. . . 4 |- (R e. Vec -> (1st` (2nd` R)) e. Grp)
54	53	3ad2ant1 897	. . 3 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (1st` (2nd` R)) e. Grp)
55	39, 4, 3	prvs 14821	. . . 4 |- ((R e. Vec /\ (~t ` A) e. X /\ U e. W) -> ((~t ` A).w U) e. W)
56	1, 47, 37, 55	syl111anc 1100	. . 3 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((~t ` A).w U) e. W)
57	4, 42	claddinvvec 14803	. . . 4 |- ((R e. Vec /\ (A.w U) e. W) -> (~w ` (A.w U)) e. W)
58	1, 9, 57	syl11anc 524	. . 3 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (~w ` (A.w U)) e. W)
59	4	grplcan 9359	. . 3 |- (((1st` (2nd` R)) e. Grp /\ (((~t ` A).w U) e. W /\ (~w ` (A.w U)) e. W /\ (A.w U) e. W)) -> (((A.w U)(1st` (2nd` R))((~t ` A).w U)) = ((A.w U)(1st` (2nd` R))(~w ` (A.w U))) <-> ((~t ` A).w U) = (~w ` (A.w U))))
60	54, 56, 58, 9, 59	syl13anc 1102	. 2 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> (((A.w U)(1st` (2nd` R))((~t ` A).w U)) = ((A.w U)(1st` (2nd` R))(~w ` (A.w U))) <-> ((~t ` A).w U) = (~w ` (A.w U))))
61	50, 60	mpbid 212	1 |- ((R e. Vec /\ <.+t , .t >. e. Ring /\ (A e. X /\ U e. W)) -> ((~t ` A).w U) = (~w ` (A.w U)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  Idcgi 9312  invcgn 9313   /g cgs 9314  Abelcabl 9407  Ringcring 9463  Veccvec 14792

This theorem is referenced by:  muldisc 14824

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

Copyright terms: Public domain