Step |
Hyp |
Ref |
Expression |
1 |
|
recnaddnred.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
recnaddnred.b |
⊢ ( 𝜑 → 𝐵 ∈ ( ℂ ∖ ℝ ) ) |
3 |
2
|
eldifbd |
⊢ ( 𝜑 → ¬ 𝐵 ∈ ℝ ) |
4 |
|
df-nel |
⊢ ( ( 𝐴 + 𝐵 ) ∉ ℝ ↔ ¬ ( 𝐴 + 𝐵 ) ∈ ℝ ) |
5 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
6 |
2
|
eldifad |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
7 |
5 6
|
addcld |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℂ ) |
8 |
|
reim0b |
⊢ ( ( 𝐴 + 𝐵 ) ∈ ℂ → ( ( 𝐴 + 𝐵 ) ∈ ℝ ↔ ( ℑ ‘ ( 𝐴 + 𝐵 ) ) = 0 ) ) |
9 |
7 8
|
syl |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) ∈ ℝ ↔ ( ℑ ‘ ( 𝐴 + 𝐵 ) ) = 0 ) ) |
10 |
1
|
reim0d |
⊢ ( 𝜑 → ( ℑ ‘ 𝐴 ) = 0 ) |
11 |
10
|
oveq1d |
⊢ ( 𝜑 → ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ 𝐵 ) ) = ( 0 + ( ℑ ‘ 𝐵 ) ) ) |
12 |
6
|
imcld |
⊢ ( 𝜑 → ( ℑ ‘ 𝐵 ) ∈ ℝ ) |
13 |
12
|
recnd |
⊢ ( 𝜑 → ( ℑ ‘ 𝐵 ) ∈ ℂ ) |
14 |
13
|
addid2d |
⊢ ( 𝜑 → ( 0 + ( ℑ ‘ 𝐵 ) ) = ( ℑ ‘ 𝐵 ) ) |
15 |
11 14
|
eqtrd |
⊢ ( 𝜑 → ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ 𝐵 ) ) = ( ℑ ‘ 𝐵 ) ) |
16 |
15
|
eqeq1d |
⊢ ( 𝜑 → ( ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ 𝐵 ) ) = 0 ↔ ( ℑ ‘ 𝐵 ) = 0 ) ) |
17 |
5 6
|
imaddd |
⊢ ( 𝜑 → ( ℑ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ 𝐵 ) ) ) |
18 |
17
|
eqeq1d |
⊢ ( 𝜑 → ( ( ℑ ‘ ( 𝐴 + 𝐵 ) ) = 0 ↔ ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ 𝐵 ) ) = 0 ) ) |
19 |
|
reim0b |
⊢ ( 𝐵 ∈ ℂ → ( 𝐵 ∈ ℝ ↔ ( ℑ ‘ 𝐵 ) = 0 ) ) |
20 |
6 19
|
syl |
⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ↔ ( ℑ ‘ 𝐵 ) = 0 ) ) |
21 |
16 18 20
|
3bitr4d |
⊢ ( 𝜑 → ( ( ℑ ‘ ( 𝐴 + 𝐵 ) ) = 0 ↔ 𝐵 ∈ ℝ ) ) |
22 |
9 21
|
bitrd |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) ∈ ℝ ↔ 𝐵 ∈ ℝ ) ) |
23 |
22
|
notbid |
⊢ ( 𝜑 → ( ¬ ( 𝐴 + 𝐵 ) ∈ ℝ ↔ ¬ 𝐵 ∈ ℝ ) ) |
24 |
4 23
|
syl5bb |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) ∉ ℝ ↔ ¬ 𝐵 ∈ ℝ ) ) |
25 |
3 24
|
mpbird |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∉ ℝ ) |