Theorem hmdmadj

Description: Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	hmdmadj	⊢ ( 𝑇 ∈ HrmOp → 𝑇 ∈ dom adjℎ )

Proof

Step	Hyp	Ref	Expression
1		hmopf	⊢ ( 𝑇 ∈ HrmOp → 𝑇 : ℋ ⟶ ℋ )
2		hon0	⊢ ( 𝑇 : ℋ ⟶ ℋ → ¬ 𝑇 = ∅ )
3	1 2	syl	⊢ ( 𝑇 ∈ HrmOp → ¬ 𝑇 = ∅ )
4		hmopadj	⊢ ( 𝑇 ∈ HrmOp → ( adjℎ ‘ 𝑇 ) = 𝑇 )
5	4	eqeq1d	⊢ ( 𝑇 ∈ HrmOp → ( ( adjℎ ‘ 𝑇 ) = ∅ ↔ 𝑇 = ∅ ) )
6	3 5	mtbird	⊢ ( 𝑇 ∈ HrmOp → ¬ ( adjℎ ‘ 𝑇 ) = ∅ )
7		ndmfv	⊢ ( ¬ 𝑇 ∈ dom adjℎ → ( adjℎ ‘ 𝑇 ) = ∅ )
8	6 7	nsyl2	⊢ ( 𝑇 ∈ HrmOp → 𝑇 ∈ dom adjℎ )