Theorem lmodvs0d

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

	Ref	Expression
Hypotheses	lmodvs0d.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
	lmodvs0d.k	$$⊢K={\mathrm{Base}}_F$$
	lmodvs0d.s	$$⊢\underset{\dot{}}{·}={\cdot }_W$$
	lmodvs0d.z	$$⊢\underset{\dot{}}{0}=0_W$$
	lmodvs0d.w	$$⊢\phi \to W\in \mathrm{LMod}$$
	lmodvs0d.x	$$⊢\phi \to X\in K$$
Assertion	lmodvs0d	$$⊢\phi \to X\underset{\dot{}}{·}\underset{\dot{}}{0}=\underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		lmodvs0d.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		lmodvs0d.k	$$⊢K={\mathrm{Base}}_F$$
3		lmodvs0d.s	$$⊢\underset{\dot{}}{·}={\cdot }_W$$
4		lmodvs0d.z	$$⊢\underset{\dot{}}{0}=0_W$$
5		lmodvs0d.w	$$⊢\phi \to W\in \mathrm{LMod}$$
6		lmodvs0d.x	$$⊢\phi \to X\in K$$
7	1 3 2 4	lmodvs0	$$⊢\left(W\in \mathrm{LMod}\wedge X\in K\right)\to X\underset{\dot{}}{·}\underset{\dot{}}{0}=\underset{\dot{}}{0}$$
8	5 6 7	syl2anc	$$⊢\phi \to X\underset{\dot{}}{·}\underset{\dot{}}{0}=\underset{\dot{}}{0}$$