Theorem pjcohcli

Description: Closure of composition of projections. (Contributed by NM, 7-Oct-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjco.1	$$⊢G\in {\mathbf{C}}_{\mathscr{H}}$$
	pjco.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	pjcohcli	$$⊢A\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(A\right)\in \mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		pjco.1	$$⊢G\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjco.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
3	1	pjfi	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right):\mathscr{H}⟶\mathscr{H}$$
4	2	pjfi	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right):\mathscr{H}⟶\mathscr{H}$$
5	3 4	hococli	$$⊢A\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(A\right)\in \mathscr{H}$$