Theorem pjclii

Description: Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjcli.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
	pjcli.2	$$⊢A\in \mathscr{H}$$
Assertion	pjclii	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in H$$

Proof

Step	Hyp	Ref	Expression
1		pjcli.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjcli.2	$$⊢A\in \mathscr{H}$$
3	1	pjcli	$$⊢A\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in H$$
4	2 3	ax-mp	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in H$$