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Metamath Proof Explorer

Theorem ho0f

Description: Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	ho0f	$⊢ 0_{hop} : ℋ ⟶ ℋ$

Step	Hyp	Ref	Expression
1		h0elch	$⊢ 0_{ℋ} \in C_{ℋ}$
2	1	pjfi	$⊢ {proj}_{ℎ} (0_{ℋ}) : ℋ ⟶ ℋ$
3		df-h0op	$⊢ 0_{hop} = {proj}_{ℎ} (0_{ℋ})$
4	3	feq1i	$⊢ 0_{hop} : ℋ ⟶ ℋ \leftrightarrow {proj}_{ℎ} (0_{ℋ}) : ℋ ⟶ ℋ$
5	2 4	mpbir	$⊢ 0_{hop} : ℋ ⟶ ℋ$