Theorem mptscmfsuppd

Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe . (Contributed by Stefan O'Rear, 21-Mar-2015) (Revised by AV, 8-Aug-2019) (Proof shortened by AV, 18-Oct-2019)

	Ref	Expression
Hypotheses	mptscmfsuppd.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	mptscmfsuppd.s	⊢ 𝑆 = ( Scalar ‘ 𝑃 )
	mptscmfsuppd.n	⊢ · = ( ·𝑠 ‘ 𝑃 )
	mptscmfsuppd.p	⊢ ( 𝜑 → 𝑃 ∈ LMod )
	mptscmfsuppd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	mptscmfsuppd.z	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝑍 ∈ 𝐵 )
	mptscmfsuppd.a	⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ 𝑌 )
	mptscmfsuppd.f	⊢ ( 𝜑 → 𝐴 finSupp ( 0g ‘ 𝑆 ) )
Assertion	mptscmfsuppd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑘 ) · 𝑍 ) ) finSupp ( 0g ‘ 𝑃 ) )

Proof

Step	Hyp	Ref	Expression
1		mptscmfsuppd.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
2		mptscmfsuppd.s	⊢ 𝑆 = ( Scalar ‘ 𝑃 )
3		mptscmfsuppd.n	⊢ · = ( ·𝑠 ‘ 𝑃 )
4		mptscmfsuppd.p	⊢ ( 𝜑 → 𝑃 ∈ LMod )
5		mptscmfsuppd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
6		mptscmfsuppd.z	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝑍 ∈ 𝐵 )
7		mptscmfsuppd.a	⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ 𝑌 )
8		mptscmfsuppd.f	⊢ ( 𝜑 → 𝐴 finSupp ( 0g ‘ 𝑆 ) )
9	2	a1i	⊢ ( 𝜑 → 𝑆 = ( Scalar ‘ 𝑃 ) )
10		fvexd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) ∈ V )
11		eqid	⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 )
12		eqid	⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 )
13	7	feqmptd	⊢ ( 𝜑 → 𝐴 = ( 𝑘 ∈ 𝑋 ↦ ( 𝐴 ‘ 𝑘 ) ) )
14	13 8	eqbrtrrd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 ↦ ( 𝐴 ‘ 𝑘 ) ) finSupp ( 0g ‘ 𝑆 ) )
15	5 4 9 1 10 6 11 12 3 14	mptscmfsupp0	⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑘 ) · 𝑍 ) ) finSupp ( 0g ‘ 𝑃 ) )