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
|
syl6bb |
⊢ ( 𝑇 : ℋ ⟶ ℋ → ( 𝐴 ∈ ( eigvec ‘ 𝑇 ) ↔ ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃ 𝑥 ∈ ℂ ( 𝑇 ‘ 𝐴 ) = ( 𝑥 ·ℎ 𝐴 ) ) ) ) |