Metamath Proof Explorer

Theorem leopsq

Description: The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion leopsq ( 𝑇 ∈ HrmOp → 0hop ≤op ( 𝑇 ∘ 𝑇 ) )

Proof

Step Hyp Ref Expression
1 hmopf ⊢ ( 𝑇 ∈ HrmOp → 𝑇 : ℋ ⟶ ℋ )
2 1 ffvelrnda ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( 𝑇 ‘ 𝑥 ) ∈ ℋ )
3 hiidge0 ⊢ ( ( 𝑇 ‘ 𝑥 ) ∈ ℋ → 0 ≤ ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑥 ) ) )
4 2 3 syl ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → 0 ≤ ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑥 ) ) )
5 simpl ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → 𝑇 ∈ HrmOp )
6 simpr ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → 𝑥 ∈ ℋ )
7 hmop ⊢ ( ( 𝑇 ∈ HrmOp ∧ ( 𝑇 ‘ 𝑥 ) ∈ ℋ ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑥 ) ) = ( ( 𝑇 ‘ ( 𝑇 ‘ 𝑥 ) ) ·ih 𝑥 ) )
8 5 2 6 7 syl3anc ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑥 ) ) = ( ( 𝑇 ‘ ( 𝑇 ‘ 𝑥 ) ) ·ih 𝑥 ) )
9 fvco3 ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ∘ 𝑇 ) ‘ 𝑥 ) = ( 𝑇 ‘ ( 𝑇 ‘ 𝑥 ) ) )
10 1 9 sylan ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ∘ 𝑇 ) ‘ 𝑥 ) = ( 𝑇 ‘ ( 𝑇 ‘ 𝑥 ) ) )
11 10 oveq1d ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( ( 𝑇 ∘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) = ( ( 𝑇 ‘ ( 𝑇 ‘ 𝑥 ) ) ·ih 𝑥 ) )
12 8 11 eqtr4d ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑥 ) ) = ( ( ( 𝑇 ∘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) )
13 4 12 breqtrd ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → 0 ≤ ( ( ( 𝑇 ∘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) )
14 13 ralrimiva ⊢ ( 𝑇 ∈ HrmOp → ∀ 𝑥 ∈ ℋ 0 ≤ ( ( ( 𝑇 ∘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) )
15 eqid ⊢ ( 𝑇 ∘ 𝑇 ) = ( 𝑇 ∘ 𝑇 )
16 hmopco ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ ( 𝑇 ∘ 𝑇 ) = ( 𝑇 ∘ 𝑇 ) ) → ( 𝑇 ∘ 𝑇 ) ∈ HrmOp )
17 15 16 mp3an3 ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ) → ( 𝑇 ∘ 𝑇 ) ∈ HrmOp )
18 17 anidms ⊢ ( 𝑇 ∈ HrmOp → ( 𝑇 ∘ 𝑇 ) ∈ HrmOp )
19 leoppos ⊢ ( ( 𝑇 ∘ 𝑇 ) ∈ HrmOp → ( 0hop ≤op ( 𝑇 ∘ 𝑇 ) ↔ ∀ 𝑥 ∈ ℋ 0 ≤ ( ( ( 𝑇 ∘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) ) )
20 18 19 syl ⊢ ( 𝑇 ∈ HrmOp → ( 0hop ≤op ( 𝑇 ∘ 𝑇 ) ↔ ∀ 𝑥 ∈ ℋ 0 ≤ ( ( ( 𝑇 ∘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) ) )
21 14 20 mpbird ⊢ ( 𝑇 ∈ HrmOp → 0hop ≤op ( 𝑇 ∘ 𝑇 ) )