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Metamath Proof Explorer

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 )