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 ∈ ran projℎ → ( 𝑇 ∘ 𝑇 ) = 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		dfpjop	⊢ ( 𝑇 ∈ ran projℎ ↔ ( 𝑇 ∈ HrmOp ∧ ( 𝑇 ∘ 𝑇 ) = 𝑇 ) )
2	1	simprbi	⊢ ( 𝑇 ∈ ran projℎ → ( 𝑇 ∘ 𝑇 ) = 𝑇 )