Theorem lnopf

Description: A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	lnopf	⊢ ( 𝑇 ∈ LinOp → 𝑇 : ℋ ⟶ ℋ )

Proof

Step	Hyp	Ref	Expression
1		ellnop	⊢ ( 𝑇 ∈ LinOp ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℋ ∀ 𝑧 ∈ ℋ ( 𝑇 ‘ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( ( 𝑥 ·ℎ ( 𝑇 ‘ 𝑦 ) ) +ℎ ( 𝑇 ‘ 𝑧 ) ) ) )
2	1	simplbi	⊢ ( 𝑇 ∈ LinOp → 𝑇 : ℋ ⟶ ℋ )