Theorem hoival

Description: The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	hoival	⊢ ( 𝐴 ∈ ℋ → ( Iop ‘ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		df-iop	⊢ Iop = ( projℎ ‘ ℋ )
2	1	fveq1i	⊢ ( Iop ‘ 𝐴 ) = ( ( projℎ ‘ ℋ ) ‘ 𝐴 )
3		pjch1	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) = 𝐴 )
4	2 3	eqtrid	⊢ ( 𝐴 ∈ ℋ → ( Iop ‘ 𝐴 ) = 𝐴 )