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Metamath Proof Explorer

Theorem lmod1cl

Description: The ring unity in a left module belongs to the set of scalars. (Contributed by NM, 11-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmod1cl.f	$⊢ F = Scalar (W)$
	lmod1cl.k	$⊢ K = {Base}_{F}$
	lmod1cl.u	$⊢ \underset{˙}{1} = 1_{F}$
Assertion	lmod1cl	$⊢ W \in LMod \to \underset{˙}{1} \in K$

Proof

Step	Hyp	Ref	Expression
1		lmod1cl.f	$⊢ F = Scalar (W)$
2		lmod1cl.k	$⊢ K = {Base}_{F}$
3		lmod1cl.u	$⊢ \underset{˙}{1} = 1_{F}$
4	1	lmodring	$⊢ W \in LMod \to F \in Ring$
5	2 3	ringidcl	$⊢ F \in Ring \to \underset{˙}{1} \in K$
6	4 5	syl	$⊢ W \in LMod \to \underset{˙}{1} \in K$