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 ) |
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 ) |