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Metamath Proof Explorer

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 ) )