Theorem ressip

Description: The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Hypotheses	resssca.1	$$⊢H=G{\upharpoonright }_{𝑠}A$$
	ressip.2	$$⊢\underset{\dot{}}{,}={\cdot }_{𝑖}\left(G\right)$$
Assertion	ressip	$$⊢A\in V\to \underset{\dot{}}{,}={\cdot }_{𝑖}\left(H\right)$$

Proof

Step	Hyp	Ref	Expression
1		resssca.1	$$⊢H=G{\upharpoonright }_{𝑠}A$$
2		ressip.2	$$⊢\underset{\dot{}}{,}={\cdot }_{𝑖}\left(G\right)$$
3		ipid	$$⊢{\cdot }_{𝑖}=\mathrm{Slot}{\cdot }_{𝑖}\left(\mathrm{ndx}\right)$$
4		ipndxnbasendx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
5	1 2 3 4	resseqnbas	$$⊢A\in V\to \underset{\dot{}}{,}={\cdot }_{𝑖}\left(H\right)$$