Theorem resvmulr

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

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

Proof

Step	Hyp	Ref	Expression
1		resvbas.1	⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
2		resvmulr.2	⊢ · = ( .r ‘ 𝐺 )
3		mulrid	⊢ .r = Slot ( .r ‘ ndx )
4		scandxnmulrndx	⊢ ( Scalar ‘ ndx ) ≠ ( .r ‘ ndx )
5	4	necomi	⊢ ( .r ‘ ndx ) ≠ ( Scalar ‘ ndx )
6	1 2 3 5	resvlem	⊢ ( 𝐴 ∈ 𝑉 → · = ( .r ‘ 𝐻 ) )