Metamath Proof Explorer


Theorem slmd0vlid

Description: Left identity law for the zero vector. ( hvaddid2 analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses slmd0vlid.v
|- V = ( Base ` W )
slmd0vlid.a
|- .+ = ( +g ` W )
slmd0vlid.z
|- .0. = ( 0g ` W )
Assertion slmd0vlid
|- ( ( W e. SLMod /\ X e. V ) -> ( .0. .+ X ) = X )

Proof

Step Hyp Ref Expression
1 slmd0vlid.v
 |-  V = ( Base ` W )
2 slmd0vlid.a
 |-  .+ = ( +g ` W )
3 slmd0vlid.z
 |-  .0. = ( 0g ` W )
4 slmdmnd
 |-  ( W e. SLMod -> W e. Mnd )
5 1 2 3 mndlid
 |-  ( ( W e. Mnd /\ X e. V ) -> ( .0. .+ X ) = X )
6 4 5 sylan
 |-  ( ( W e. SLMod /\ X e. V ) -> ( .0. .+ X ) = X )