Theorem 0leopj

Description: A projector is a positive operator. (Contributed by NM, 27-Sep-2008) (New usage is discouraged.)

	Ref	Expression
Assertion	0leopj	⊢ ( 𝑇 ∈ ran projℎ → 0hop ≤op 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		elpjhmop	⊢ ( 𝑇 ∈ ran projℎ → 𝑇 ∈ HrmOp )
2		leopsq	⊢ ( 𝑇 ∈ HrmOp → 0hop ≤op ( 𝑇 ∘ 𝑇 ) )
3	1 2	syl	⊢ ( 𝑇 ∈ ran projℎ → 0hop ≤op ( 𝑇 ∘ 𝑇 ) )
4		elpjidm	⊢ ( 𝑇 ∈ ran projℎ → ( 𝑇 ∘ 𝑇 ) = 𝑇 )
5	3 4	breqtrd	⊢ ( 𝑇 ∈ ran projℎ → 0hop ≤op 𝑇 )