Theorem lmodvpncan

Description: Addition/subtraction cancellation law for vectors. ( hvpncan analog.) (Contributed by NM, 16-Apr-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmod4.v	|- V = ( Base ` W )
	lmod4.p	|- .+ = ( +g ` W )
	lmodvaddsub4.m	|- .- = ( -g ` W )
Assertion	lmodvpncan	|- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .+ B ) .- B ) = A )

Proof

Step	Hyp	Ref	Expression
1		lmod4.v	|- V = ( Base ` W )
2		lmod4.p	|- .+ = ( +g ` W )
3		lmodvaddsub4.m	|- .- = ( -g ` W )
4		lmodgrp	|- ( W e. LMod -> W e. Grp )
5	1 2 3	grppncan	|- ( ( W e. Grp /\ A e. V /\ B e. V ) -> ( ( A .+ B ) .- B ) = A )
6	4 5	syl3an1	|- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .+ B ) .- B ) = A )