Theorem elpjidm

Description: A projection operator is idempotent. Part of Theorem 26.1 of Halmos p. 43. (Contributed by NM, 24-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	elpjidm	$$⊢T\in \mathrm{ran}{\mathrm{proj}}_{ℎ}\to T\circ T=T$$

Proof

Step	Hyp	Ref	Expression
1		dfpjop	$$⊢T\in \mathrm{ran}{\mathrm{proj}}_{ℎ}\leftrightarrow \left(T\in \mathrm{HrmOp}\wedge T\circ T=T\right)$$
2	1	simprbi	$$⊢T\in \mathrm{ran}{\mathrm{proj}}_{ℎ}\to T\circ T=T$$