Theorem dmadjop

Description: A member of the domain of the adjoint function is a Hilbert space operator. (Contributed by NM, 15-Feb-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	dmadjop	⊢ ( 𝑇 ∈ dom adjℎ → 𝑇 : ℋ ⟶ ℋ )

Proof

Step	Hyp	Ref	Expression
1		dmadjss	⊢ dom adjℎ ⊆ ( ℋ ↑m ℋ )
2	1	sseli	⊢ ( 𝑇 ∈ dom adjℎ → 𝑇 ∈ ( ℋ ↑m ℋ ) )
3		ax-hilex	⊢ ℋ ∈ V
4	3 3	elmap	⊢ ( 𝑇 ∈ ( ℋ ↑m ℋ ) ↔ 𝑇 : ℋ ⟶ ℋ )
5	2 4	sylib	⊢ ( 𝑇 ∈ dom adjℎ → 𝑇 : ℋ ⟶ ℋ )