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Metamath Proof Explorer

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 → ( I_op ‘ 𝐴 ) = 𝐴 )

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