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 ) )