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Metamath Proof Explorer

Theorem eighmre

Description: The eigenvalues of a Hermitian operator are real. Equation 1.30 of Hughes p. 49. (Contributed by NM, 19-Mar-2006) (New usage is discouraged.)

Ref Expression
Assertion eighmre ( ( 𝑇 ∈ HrmOp ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 hmopf ⊢ ( 𝑇 ∈ HrmOp → 𝑇 : ℋ ⟶ ℋ )
2 eleigveccl ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → 𝐴 ∈ ℋ )
3 eigvalcl ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℂ )
4 2 3 jca ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( 𝐴 ∈ ℋ ∧ ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℂ ) )
5 eigvec1 ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( ( 𝑇 ‘ 𝐴 ) = ( ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) )
6 4 5 jca ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( ( 𝐴 ∈ ℋ ∧ ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℂ ) ∧ ( ( 𝑇 ‘ 𝐴 ) = ( ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) ) )
7 1 6 sylan ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( ( 𝐴 ∈ ℋ ∧ ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℂ ) ∧ ( ( 𝑇 ‘ 𝐴 ) = ( ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) ) )
8 2 2 jca ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) )
9 1 8 sylan ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) )
10 hmop ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) )
11 10 3expb ⊢ ( ( 𝑇 ∈ HrmOp ∧ ( 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) ) → ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) )
12 9 11 syldan ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) )
13 eigre ⊢ ( ( ( 𝐴 ∈ ℋ ∧ ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℂ ) ∧ ( ( 𝑇 ‘ 𝐴 ) = ( ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) ) → ( ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) ↔ ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℝ ) )
14 13 biimpa ⊢ ( ( ( ( 𝐴 ∈ ℋ ∧ ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℂ ) ∧ ( ( 𝑇 ‘ 𝐴 ) = ( ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) ) ∧ ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) ) → ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℝ )
15 7 12 14 syl2anc ⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝐴 ∈ ( eigvec ‘ 𝑇 ) ) → ( ( eigval ‘ 𝑇 ) ‘ 𝐴 ) ∈ ℝ )