Step |
Hyp |
Ref |
Expression |
1 |
|
recnaddnred.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
recnaddnred.b |
⊢ ( 𝜑 → 𝐵 ∈ ( ℂ ∖ ℝ ) ) |
3 |
|
cndivrenred.n |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
4 |
2
|
eldifbd |
⊢ ( 𝜑 → ¬ 𝐵 ∈ ℝ ) |
5 |
|
df-nel |
⊢ ( ( 𝐵 / 𝐴 ) ∉ ℝ ↔ ¬ ( 𝐵 / 𝐴 ) ∈ ℝ ) |
6 |
2
|
eldifad |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
7 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
8 |
6 7 3
|
divcld |
⊢ ( 𝜑 → ( 𝐵 / 𝐴 ) ∈ ℂ ) |
9 |
|
reim0b |
⊢ ( ( 𝐵 / 𝐴 ) ∈ ℂ → ( ( 𝐵 / 𝐴 ) ∈ ℝ ↔ ( ℑ ‘ ( 𝐵 / 𝐴 ) ) = 0 ) ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → ( ( 𝐵 / 𝐴 ) ∈ ℝ ↔ ( ℑ ‘ ( 𝐵 / 𝐴 ) ) = 0 ) ) |
11 |
6
|
imcld |
⊢ ( 𝜑 → ( ℑ ‘ 𝐵 ) ∈ ℝ ) |
12 |
11
|
recnd |
⊢ ( 𝜑 → ( ℑ ‘ 𝐵 ) ∈ ℂ ) |
13 |
12 7 3
|
diveq0ad |
⊢ ( 𝜑 → ( ( ( ℑ ‘ 𝐵 ) / 𝐴 ) = 0 ↔ ( ℑ ‘ 𝐵 ) = 0 ) ) |
14 |
1 6 3
|
imdivd |
⊢ ( 𝜑 → ( ℑ ‘ ( 𝐵 / 𝐴 ) ) = ( ( ℑ ‘ 𝐵 ) / 𝐴 ) ) |
15 |
14
|
eqeq1d |
⊢ ( 𝜑 → ( ( ℑ ‘ ( 𝐵 / 𝐴 ) ) = 0 ↔ ( ( ℑ ‘ 𝐵 ) / 𝐴 ) = 0 ) ) |
16 |
|
reim0b |
⊢ ( 𝐵 ∈ ℂ → ( 𝐵 ∈ ℝ ↔ ( ℑ ‘ 𝐵 ) = 0 ) ) |
17 |
6 16
|
syl |
⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ↔ ( ℑ ‘ 𝐵 ) = 0 ) ) |
18 |
13 15 17
|
3bitr4d |
⊢ ( 𝜑 → ( ( ℑ ‘ ( 𝐵 / 𝐴 ) ) = 0 ↔ 𝐵 ∈ ℝ ) ) |
19 |
10 18
|
bitrd |
⊢ ( 𝜑 → ( ( 𝐵 / 𝐴 ) ∈ ℝ ↔ 𝐵 ∈ ℝ ) ) |
20 |
19
|
notbid |
⊢ ( 𝜑 → ( ¬ ( 𝐵 / 𝐴 ) ∈ ℝ ↔ ¬ 𝐵 ∈ ℝ ) ) |
21 |
5 20
|
syl5bb |
⊢ ( 𝜑 → ( ( 𝐵 / 𝐴 ) ∉ ℝ ↔ ¬ 𝐵 ∈ ℝ ) ) |
22 |
4 21
|
mpbird |
⊢ ( 𝜑 → ( 𝐵 / 𝐴 ) ∉ ℝ ) |