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		df-ip	$$⊢{\cdot }_{𝑖}=\mathrm{Slot}8$$
4		8nn	$$⊢8\in \mathbb{N}$$
5		1lt8	$$⊢1<8$$
6	1 2 3 4 5	resslem	$$⊢A\in V\to \underset{\dot{}}{,}={\cdot }_{𝑖}\left(H\right)$$