Theorem resvbas

Description: Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018) (Revised by AV, 31-Oct-2024)

	Ref	Expression
Hypotheses	resvbas.1	$$⊢H=G{\upharpoonright }_{𝑣}A$$
	resvbas.2	$$⊢B={\mathrm{Base}}_G$$
Assertion	resvbas	$$⊢A\in V\to B={\mathrm{Base}}_H$$

Proof

Step	Hyp	Ref	Expression
1		resvbas.1	$$⊢H=G{\upharpoonright }_{𝑣}A$$
2		resvbas.2	$$⊢B={\mathrm{Base}}_G$$
3		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
4		scandxnbasendx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
5	4	necomi	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)$$
6	1 2 3 5	resvlem	$$⊢A\in V\to B={\mathrm{Base}}_H$$