Metamath Proof Explorer


Theorem lmod0vlid

Description: Left identity law for the zero vector. ( hvaddid2 analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses 0vlid.v
|- V = ( Base ` W )
0vlid.a
|- .+ = ( +g ` W )
0vlid.z
|- .0. = ( 0g ` W )
Assertion lmod0vlid
|- ( ( W e. LMod /\ X e. V ) -> ( .0. .+ X ) = X )

Proof

Step Hyp Ref Expression
1 0vlid.v
 |-  V = ( Base ` W )
2 0vlid.a
 |-  .+ = ( +g ` W )
3 0vlid.z
 |-  .0. = ( 0g ` W )
4 lmodgrp
 |-  ( W e. LMod -> W e. Grp )
5 1 2 3 grplid
 |-  ( ( W e. Grp /\ X e. V ) -> ( .0. .+ X ) = X )
6 4 5 sylan
 |-  ( ( W e. LMod /\ X e. V ) -> ( .0. .+ X ) = X )