Metamath Proof Explorer


Theorem slmd1cl

Description: The ring unit in a semiring left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses slmd1cl.f F = Scalar W
slmd1cl.k K = Base F
slmd1cl.u 1 ˙ = 1 F
Assertion slmd1cl W SLMod 1 ˙ K

Proof

Step Hyp Ref Expression
1 slmd1cl.f F = Scalar W
2 slmd1cl.k K = Base F
3 slmd1cl.u 1 ˙ = 1 F
4 1 slmdsrg W SLMod F SRing
5 2 3 srgidcl F SRing 1 ˙ K
6 4 5 syl W SLMod 1 ˙ K