Theorem lmodsubeq0

Description: If the difference between two vectors is zero, they are equal. ( hvsubeq0 analog.) (Contributed by NM, 31-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodsubeq0.v	|- V = ( Base ` W )
	lmodsubeq0.o	|- .0. = ( 0g ` W )
	lmodsubeq0.m	|- .- = ( -g ` W )
Assertion	lmodsubeq0	|- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .- B ) = .0. <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		lmodsubeq0.v	|- V = ( Base ` W )
2		lmodsubeq0.o	|- .0. = ( 0g ` W )
3		lmodsubeq0.m	|- .- = ( -g ` W )
4		lmodgrp	|- ( W e. LMod -> W e. Grp )
5	1 2 3	grpsubeq0	|- ( ( W e. Grp /\ A e. V /\ B e. V ) -> ( ( A .- B ) = .0. <-> A = B ) )
6	4 5	syl3an1	|- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( ( A .- B ) = .0. <-> A = B ) )