Metamath Proof Explorer

Theorem lmod0vcl

Description: The zero vector is a vector. ( ax-hv0cl analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses 0vcl.v 
|- V = ( Base ` W )
0vcl.z 
|- .0. = ( 0g ` W )
Assertion lmod0vcl 
|- ( W e. LMod -> .0. e. V )

Proof

Step Hyp Ref Expression
1 0vcl.v 
 |-  V = ( Base ` W )
2 0vcl.z 
 |-  .0. = ( 0g ` W )
3 lmodgrp 
 |-  ( W e. LMod -> W e. Grp )
4 1 2 grpidcl 
 |-  ( W e. Grp -> .0. e. V )
5 3 4 syl 
 |-  ( W e. LMod -> .0. e. V )