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	|- .x. = ( .r ` G )
Assertion	resvmulr	|- ( A e. V -> .x. = ( .r ` H ) )

Proof

Step	Hyp	Ref	Expression
1		resvbas.1	|- H = ( G |`v A )
2		resvmulr.2	|- .x. = ( .r ` G )
3		mulridx	|- .r = Slot ( .r ` ndx )
4		scandxnmulrndx	|- ( Scalar ` ndx ) =/= ( .r ` ndx )
5	4	necomi	|- ( .r ` ndx ) =/= ( Scalar ` ndx )
6	1 2 3 5	resvlem	|- ( A e. V -> .x. = ( .r ` H ) )