Theorem resvvsca

Description: .s is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018)

	Ref	Expression
Hypotheses	resvbas.1	$$⊢H=G{\upharpoonright }_{𝑣}A$$
	resvvsca.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
Assertion	resvvsca	$$⊢A\in V\to \underset{\dot{}}{·}={\cdot }_H$$

Proof

Step	Hyp	Ref	Expression
1		resvbas.1	$$⊢H=G{\upharpoonright }_{𝑣}A$$
2		resvvsca.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
3		df-vsca	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}6$$
4		6nn	$$⊢6\in \mathbb{N}$$
5		5re	$$⊢5\in ℝ$$
6		5lt6	$$⊢5<6$$
7	5 6	gtneii	$$⊢6\ne 5$$
8	1 2 3 4 7	resvlem	$$⊢A\in V\to \underset{\dot{}}{·}={\cdot }_H$$