Theorem lmodvs0d

Description: A scalar times the zero vector is the zero vector. (Contributed by SN, 24-Sep-2026)

	Ref	Expression
Hypotheses	lmodvs0d.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	lmodvs0d.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
	lmodvs0d.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
	lmodvs0d.z	⊢ 0 = ( 0g ‘ 𝑊 )
	lmodvs0d.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	lmodvs0d.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
Assertion	lmodvs0d	⊢ ( 𝜑 → ( 𝑋 · 0 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		lmodvs0d.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		lmodvs0d.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3		lmodvs0d.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
4		lmodvs0d.z	⊢ 0 = ( 0g ‘ 𝑊 )
5		lmodvs0d.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
6		lmodvs0d.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
7	1 3 2 4	lmodvs0	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ) → ( 𝑋 · 0 ) = 0 )
8	5 6 7	syl2anc	⊢ ( 𝜑 → ( 𝑋 · 0 ) = 0 )