Theorem pjidmi

Description: A projection is idempotent. Property (ii) of Beran p. 109. (Contributed by NM, 28-Oct-1999) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		pjidm.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjidm.2	$$⊢A\in \mathscr{H}$$
3	1 2	pjclii	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in H$$
4	1 2	pjhclii	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in \mathscr{H}$$
5	1 4	pjchi	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in H\leftrightarrow {\mathrm{proj}}_{ℎ}\left(H\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)={\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)$$
6	3 5	mpbi	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)={\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)$$