Theorem mvecrtol 14816

Description: Moving a vector from the right member of an equation into the left member.

Hypotheses

Ref	Expression
vwit.1	|- 0w = (Id` +w )
vwit.2	|- +w = (1st` (2nd` R))
vwit.3	|- -w = ( /g ` +w )
vwit.4	|- W = ran +w

Assertion

Ref	Expression
mvecrtol	|- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> (V1 = V2 <-> (V1-w V2) = 0w ))

Proof of Theorem mvecrtol

Step	Hyp	Ref	Expression
1		vwit.2	. . . . . 6 |- +w = (1st` (2nd` R))
2	1	vecax1 14796	. . . . 5 |- (R e. Vec -> +w e. Abel)
3		ablgrp 9410	. . . . 5 |- (+w e. Abel -> +w e. Grp)
4	2, 3	syl 12	. . . 4 |- (R e. Vec -> +w e. Grp)
5	4	adantr 425	. . 3 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> +w e. Grp)
6		simprl 450	. . 3 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> V1 e. W)
7		simprr 451	. . 3 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> V2 e. W)
8		vwit.4	. . . 4 |- W = ran +w
9		vwit.3	. . . 4 |- -w = ( /g ` +w )
10	8, 9	grpdrcan 14738	. . 3 |- ((+w e. Grp /\ (V1 e. W /\ V2 e. W /\ V2 e. W)) -> ((V1-w V2) = (V2-w V2) <-> V1 = V2))
11	5, 6, 7, 7, 10	syl13anc 1102	. 2 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> ((V1-w V2) = (V2-w V2) <-> V1 = V2))
12		vwit.1	. . . . 5 |- 0w = (Id` +w )
13	12, 1, 9, 8	vwit 14814	. . . 4 |- ((R e. Vec /\ V2 e. W) -> (V2-w V2) = 0w )
14	13	adantrl 430	. . 3 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> (V2-w V2) = 0w )
15	14	eqeq2d 1895	. 2 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> ((V1-w V2) = (V2-w V2) <-> (V1-w V2) = 0w ))
16	11, 15	bitr3d 589	1 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> (V1 = V2 <-> (V1-w V2) = 0w ))

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

This theorem is referenced by:  svli2 14826

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-vec 14793

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