Theorem hmopbdoptHIL

Description: A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem). (Contributed by NM, 18-Jan-2008) (New usage is discouraged.)

	Ref	Expression
Assertion	hmopbdoptHIL	⊢ ( 𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp )

Proof

Step	Hyp	Ref	Expression
1		hmoplin	⊢ ( 𝑇 ∈ HrmOp → 𝑇 ∈ LinOp )
2		hmop	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) )
3	2	3expib	⊢ ( 𝑇 ∈ HrmOp → ( ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ) )
4	3	ralrimivv	⊢ ( 𝑇 ∈ HrmOp → ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) )
5		hilhl	⊢ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ ∈ CHilOLD
6		df-hba	⊢ ℋ = ( BaseSet ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
7		eqid	⊢ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
8	7	hhip	⊢ ·ih = ( ·𝑖OLD ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
9		eqid	⊢ ( ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ LnOp ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ ) = ( ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ LnOp ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
10	7 9	hhlnoi	⊢ LinOp = ( ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ LnOp ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
11		eqid	⊢ ( ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ BLnOp ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ ) = ( ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ BLnOp ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
12	7 11	hhbloi	⊢ BndLinOp = ( ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ BLnOp ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
13	6 8 10 12	htth	⊢ ( ( ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ ∈ CHilOLD ∧ 𝑇 ∈ LinOp ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ) → 𝑇 ∈ BndLinOp )
14	5 13	mp3an1	⊢ ( ( 𝑇 ∈ LinOp ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ) → 𝑇 ∈ BndLinOp )
15	1 4 14	syl2anc	⊢ ( 𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp )