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	$$⊢H=G{\upharpoonright }_{𝑣}A$$
	resvmulr.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
Assertion	resvmulr	$$⊢A\in V\to \underset{\dot{}}{·}={\cdot }_H$$

Proof

Step	Hyp	Ref	Expression
1		resvbas.1	$$⊢H=G{\upharpoonright }_{𝑣}A$$
2		resvmulr.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
3		mulrid	$$⊢{\cdot }_{𝑟}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		scandxnmulrndx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$
5	4	necomi	$$⊢{\cdot }_{\mathrm{ndx}}\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)$$
6	1 2 3 5	resvlem	$$⊢A\in V\to \underset{\dot{}}{·}={\cdot }_H$$