Theorem invaddvec 14810

Description: Additive inverse of a sum of vectors.

Hypotheses

Ref	Expression
sum2vv.1	|- +w = (1st` (2nd` R))
sum2vv.2	|- W = ran +w
invaddvec.2	|- ~w = (inv` +w )

Assertion

Ref	Expression
invaddvec	|- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> (~w ` (V1+w V2)) = ((~w ` V1)+w (~w ` V2)))

Proof of Theorem invaddvec

Step	Hyp	Ref	Expression
1		sum2vv.1	. . . . 5 |- +w = (1st` (2nd` R))
2	1	vecax1 14796	. . . 4 |- (R e. Vec -> +w e. Abel)
3		ablgrp 9410	. . . 4 |- (+w e. Abel -> +w e. Grp)
4		sum2vv.2	. . . . . 6 |- W = ran +w
5		invaddvec.2	. . . . . 6 |- ~w = (inv` +w )
6	4, 5	grpinvop 9365	. . . . 5 |- ((+w e. Grp /\ V1 e. W /\ V2 e. W) -> (~w ` (V1+w V2)) = ((~w ` V2)+w (~w ` V1)))
7	6	3expib 1070	. . . 4 |- (+w e. Grp -> ((V1 e. W /\ V2 e. W) -> (~w ` (V1+w V2)) = ((~w ` V2)+w (~w ` V1))))
8	2, 3, 7	3syl 24	. . 3 |- (R e. Vec -> ((V1 e. W /\ V2 e. W) -> (~w ` (V1+w V2)) = ((~w ` V2)+w (~w ` V1))))
9	8	imp 377	. 2 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> (~w ` (V1+w V2)) = ((~w ` V2)+w (~w ` V1)))
10	1	rneqi 4187	. . . . . . . . 9 |- ran +w = ran (1st` (2nd` R))
11	4, 10	eqtri 1908	. . . . . . . 8 |- W = ran (1st` (2nd` R))
12	1	fveq2i 4684	. . . . . . . . 9 |- (inv` +w ) = (inv` (1st` (2nd` R)))
13	5, 12	eqtri 1908	. . . . . . . 8 |- ~w = (inv` (1st` (2nd` R)))
14	11, 13	claddinvvec 14803	. . . . . . 7 |- ((R e. Vec /\ V1 e. W) -> (~w ` V1) e. W)
15	14	ex 402	. . . . . 6 |- (R e. Vec -> (V1 e. W -> (~w ` V1) e. W))
16	11, 13	claddinvvec 14803	. . . . . . 7 |- ((R e. Vec /\ V2 e. W) -> (~w ` V2) e. W)
17	16	ex 402	. . . . . 6 |- (R e. Vec -> (V2 e. W -> (~w ` V2) e. W))
18	15, 17	anim12d 617	. . . . 5 |- (R e. Vec -> ((V1 e. W /\ V2 e. W) -> ((~w ` V1) e. W /\ (~w ` V2) e. W)))
19	18	imp 377	. . . 4 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> ((~w ` V1) e. W /\ (~w ` V2) e. W))
20	19	ancomd 483	. . 3 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> ((~w ` V2) e. W /\ (~w ` V1) e. W))
21	1, 4	addvecom 14809	. . 3 |- ((R e. Vec /\ ((~w ` V2) e. W /\ (~w ` V1) e. W)) -> ((~w ` V2)+w (~w ` V1)) = ((~w ` V1)+w (~w ` V2)))
22	20, 21	syldan 516	. 2 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> ((~w ` V2)+w (~w ` V1)) = ((~w ` V1)+w (~w ` V2)))
23	9, 22	eqtrd 1925	1 |- ((R e. Vec /\ (V1 e. W /\ V2 e. W)) -> (~w ` (V1+w V2)) = ((~w ` V1)+w (~w ` V2)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  invcgn 9313  Abelcabl 9407  Veccvec 14792

This theorem is referenced by:  dblsubvec 14817

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-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-vec 14793

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