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Metamath Proof Explorer

Theorem eleigvec

Description: Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

Ref Expression
Assertion eleigvec ( 𝑇 : ℋ ⟶ ℋ → ( 𝐴 ∈ ( eigvec ‘ 𝑇 ) ↔ ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) ) )

Proof

Step Hyp Ref Expression
1 eigvecval ⊢ ( 𝑇 : ℋ ⟶ ℋ → ( eigvec ‘ 𝑇 ) = { 𝑦 ∈ ( ℋ ∖ 0ℋ ) ∣ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝑦 ) = ( 𝑥 ·ℎ 𝑦 ) } )
2 1 eleq2d ⊢ ( 𝑇 : ℋ ⟶ ℋ → ( 𝐴 ∈ ( eigvec ‘ 𝑇 ) ↔ 𝐴 ∈ { 𝑦 ∈ ( ℋ ∖ 0ℋ ) ∣ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝑦 ) = ( 𝑥 ·ℎ 𝑦 ) } ) )
3 eldif ⊢ ( 𝐴 ∈ ( ℋ ∖ 0ℋ ) ↔ ( 𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ ) )
4 elch0 ⊢ ( 𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ )
5 4 necon3bbii ⊢ ( ¬ 𝐴 ∈ 0ℋ ↔ 𝐴 ≠ 0ℎ )
6 5 anbi2i ⊢ ( ( 𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ ) ↔ ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) )
7 3 6 bitri ⊢ ( 𝐴 ∈ ( ℋ ∖ 0ℋ ) ↔ ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) )
8 7 anbi1i ⊢ ( ( 𝐴 ∈ ( ℋ ∖ 0ℋ ) ∧ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) ↔ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) )
9 fveq2 ⊢ ( 𝑦 = 𝐴 → ( 𝑇 ‘ 𝑦 ) = ( 𝑇 ‘ 𝐴 ) )
10 oveq2 ⊢ ( 𝑦 = 𝐴 → ( 𝑥 ·ℎ 𝑦 ) = ( 𝑥 ·ℎ 𝐴 ) )
11 9 10 eqeq12d ⊢ ( 𝑦 = 𝐴 → ( ( 𝑇 ‘ 𝑦 ) = ( 𝑥 ·ℎ 𝑦 ) ↔ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) )
12 11 rexbidv ⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝑦 ) = ( 𝑥 ·ℎ 𝑦 ) ↔ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) )
13 12 elrab ⊢ ( 𝐴 ∈ { 𝑦 ∈ ( ℋ ∖ 0ℋ ) ∣ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝑦 ) = ( 𝑥 ·ℎ 𝑦 ) } ↔ ( 𝐴 ∈ ( ℋ ∖ 0ℋ ) ∧ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) )
14 df-3an ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) ↔ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) )
15 8 13 14 3bitr4i ⊢ ( 𝐴 ∈ { 𝑦 ∈ ( ℋ ∖ 0ℋ ) ∣ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝑦 ) = ( 𝑥 ·ℎ 𝑦 ) } ↔ ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) )
16 2 15 bitrdi ⊢ ( 𝑇 : ℋ ⟶ ℋ → ( 𝐴 ∈ ( eigvec ‘ 𝑇 ) ↔ ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) ) )