Theorem pjidmcoi

Description: A projection is idempotent. Property (ii) of Beran p. 109. (Contributed by NM, 1-Oct-2000) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjidmco.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	pjidmcoi	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(H\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjidmco.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2		ssid	$$⊢H\subseteq H$$
3	1 1	pjss2coi	$$⊢H\subseteq H\leftrightarrow {\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(H\right)$$
4	2 3	mpbi	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)={\mathrm{proj}}_{ℎ}\left(H\right)$$