Theorem lmodsn0

Description: The set of scalars in a left module is nonempty. (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodsn0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	lmodsn0.b	⊢ 𝐵 = ( Base ‘ 𝐹 )
Assertion	lmodsn0	⊢ ( 𝑊 ∈ LMod → 𝐵 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		lmodsn0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		lmodsn0.b	⊢ 𝐵 = ( Base ‘ 𝐹 )
3	1	lmodfgrp	⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Grp )
4	2	grpbn0	⊢ ( 𝐹 ∈ Grp → 𝐵 ≠ ∅ )
5	3 4	syl	⊢ ( 𝑊 ∈ LMod → 𝐵 ≠ ∅ )