Theorem pjcli

Description: Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		pjcl.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2		axpjcl	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in H$$
3	1 2	mpan	$$⊢A\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in H$$