Theorem resvmulr

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

	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		df-mulr	⊢ .r = Slot 3
4		3nn	⊢ 3 ∈ ℕ
5		3re	⊢ 3 ∈ ℝ
6		3lt5	⊢ 3 < 5
7	5 6	ltneii	⊢ 3 ≠ 5
8	1 2 3 4 7	resvlem	⊢ ( 𝐴 ∈ 𝑉 → · = ( .r ‘ 𝐻 ) )