Theorem pjcocli

Description: Closure of composition of projections. (Contributed by NM, 29-Nov-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	pjcocli	$$⊢A\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(A\right)\in G$$

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 2	pjcoi	$$⊢A\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)$$
4	2	pjhcli	$$⊢A\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in \mathscr{H}$$
5	1	pjcli	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)\in G$$
6	4 5	syl	$$⊢A\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)\in G$$
7	3 6	eqeltrd	$$⊢A\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(A\right)\in G$$