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	ffvelcdmda	⊢ ( ( 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 ) )