Theorem resvplusg

Description: +g 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 )
	resvplusg.2	⊢ + = ( +g ‘ 𝐺 )
Assertion	resvplusg	⊢ ( 𝐴 ∈ 𝑉 → + = ( +g ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		resvbas.1	⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
2		resvplusg.2	⊢ + = ( +g ‘ 𝐺 )
3		plusgid	⊢ +g = Slot ( +g ‘ ndx )
4		scandxnplusgndx	⊢ ( Scalar ‘ ndx ) ≠ ( +g ‘ ndx )
5	4	necomi	⊢ ( +g ‘ ndx ) ≠ ( Scalar ‘ ndx )
6	1 2 3 5	resvlem	⊢ ( 𝐴 ∈ 𝑉 → + = ( +g ‘ 𝐻 ) )