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Metamath Proof Explorer

Theorem eigre

Description: A necessary and sufficient condition (that holds when T is a Hermitian operator) for an eigenvalue B to be real. Generalization of Equation 1.30 of Hughes p. 49. (Contributed by NM, 19-Mar-2006) (New usage is discouraged.)

Ref Expression
Assertion eigre ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ) ∧ ( ( 𝑇 ‘ 𝐴 ) = ( 𝐵 ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) ) → ( ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) ↔ 𝐵 ∈ ℝ ) )

Proof

Step Hyp Ref Expression
1 fveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝑇 ‘ 𝐴 ) = ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
2 oveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐵 ·ℎ 𝐴 ) = ( 𝐵 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
3 1 2 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝑇 ‘ 𝐴 ) = ( 𝐵 ·ℎ 𝐴 ) ↔ ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( 𝐵 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
4 neeq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ≠ 0ℎ ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ≠ 0ℎ ) )
5 3 4 anbi12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( 𝑇 ‘ 𝐴 ) = ( 𝐵 ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) ↔ ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( 𝐵 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ∧ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ≠ 0ℎ ) ) )
6 id ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
7 6 1 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
8 1 6 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) = ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
9 7 8 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) = ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
10 9 bibi1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) ↔ 𝐵 ∈ ℝ ) ↔ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) = ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ 𝐵 ∈ ℝ ) ) )
11 5 10 imbi12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( ( 𝑇 ‘ 𝐴 ) = ( 𝐵 ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) → ( ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) ↔ 𝐵 ∈ ℝ ) ) ↔ ( ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( 𝐵 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ∧ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ≠ 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) = ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ 𝐵 ∈ ℝ ) ) ) )
12 oveq1 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( 𝐵 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
13 12 eqeq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( 𝐵 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
14 13 anbi1d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( 𝐵 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ∧ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ≠ 0ℎ ) ↔ ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ∧ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ≠ 0ℎ ) ) )
15 eleq1 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( 𝐵 ∈ ℝ ↔ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∈ ℝ ) )
16 15 bibi2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) = ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ 𝐵 ∈ ℝ ) ↔ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) = ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∈ ℝ ) ) )
17 14 16 imbi12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( 𝐵 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ∧ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ≠ 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) = ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ 𝐵 ∈ ℝ ) ) ↔ ( ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ∧ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ≠ 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) = ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∈ ℝ ) ) ) )
18 ifhvhv0 ⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
19 0cn ⊢ 0 ∈ ℂ
20 19 elimel ⊢ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∈ ℂ
21 18 20 eigrei ⊢ ( ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ∧ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ≠ 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) = ( ( 𝑇 ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∈ ℝ ) )
22 11 17 21 dedth2h ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝑇 ‘ 𝐴 ) = ( 𝐵 ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) → ( ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) ↔ 𝐵 ∈ ℝ ) ) )
23 22 imp ⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ) ∧ ( ( 𝑇 ‘ 𝐴 ) = ( 𝐵 ·ℎ 𝐴 ) ∧ 𝐴 ≠ 0ℎ ) ) → ( ( 𝐴 ·ih ( 𝑇 ‘ 𝐴 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐴 ) ↔ 𝐵 ∈ ℝ ) )