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	$$⊢A\in \mathscr{H}\to I_{\mathrm{op}}\left(A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		df-iop	$$⊢I_{\mathrm{op}}={\mathrm{proj}}_{ℎ}\left(\mathscr{H}\right)$$
2	1	fveq1i	$$⊢I_{\mathrm{op}}\left(A\right)={\mathrm{proj}}_{ℎ}\left(\mathscr{H}\right)\left(A\right)$$
3		pjch1	$$⊢A\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(\mathscr{H}\right)\left(A\right)=A$$
4	2 3	syl5eq	$$⊢A\in \mathscr{H}\to I_{\mathrm{op}}\left(A\right)=A$$