Metamath Proof Explorer


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 + = ( +g𝑊 )
lmodvaddsub4.m = ( -g𝑊 )
Assertion lmodvpncan ( ( 𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉 ) → ( ( 𝐴 + 𝐵 ) 𝐵 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 lmod4.v 𝑉 = ( Base ‘ 𝑊 )
2 lmod4.p + = ( +g𝑊 )
3 lmodvaddsub4.m = ( -g𝑊 )
4 lmodgrp ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
5 1 2 3 grppncan ( ( 𝑊 ∈ Grp ∧ 𝐴𝑉𝐵𝑉 ) → ( ( 𝐴 + 𝐵 ) 𝐵 ) = 𝐴 )
6 4 5 syl3an1 ( ( 𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉 ) → ( ( 𝐴 + 𝐵 ) 𝐵 ) = 𝐴 )