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 ) )