Theorem ressip

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

	Ref	Expression
Hypotheses	resssca.1	⊢ 𝐻 = ( 𝐺 ↾s 𝐴 )
	ressip.2	⊢ , = ( ·𝑖 ‘ 𝐺 )
Assertion	ressip	⊢ ( 𝐴 ∈ 𝑉 → , = ( ·𝑖 ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		resssca.1	⊢ 𝐻 = ( 𝐺 ↾s 𝐴 )
2		ressip.2	⊢ , = ( ·𝑖 ‘ 𝐺 )
3		ipid	⊢ ·𝑖 = Slot ( ·𝑖 ‘ ndx )
4		ipndxnbasendx	⊢ ( ·𝑖 ‘ ndx ) ≠ ( Base ‘ ndx )
5	1 2 3 4	resseqnbas	⊢ ( 𝐴 ∈ 𝑉 → , = ( ·𝑖 ‘ 𝐻 ) )