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Metamath Proof Explorer

Theorem resvvsca

Description: .s is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018) (Proof shortened by AV, 31-Oct-2024)

Ref Expression
Hypotheses resvbas.1 ⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
resvvsca.2 ⊢ · = ( ·𝑠 ‘ 𝐺 )
Assertion resvvsca ( 𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘ 𝐻 ) )

Proof

Step Hyp Ref Expression
1 resvbas.1 ⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
2 resvvsca.2 ⊢ · = ( ·𝑠 ‘ 𝐺 )
3 vscaid ⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx )
4 vscandxnscandx ⊢ ( ·𝑠 ‘ ndx ) ≠ ( Scalar ‘ ndx )
5 1 2 3 4 resvlem ⊢ ( 𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘ 𝐻 ) )