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 |
⊢ ( ( 𝐴 · 𝐵 ) ∉ ℝ ↔ ¬ ( 𝐴 · 𝐵 ) ∈ ℝ ) |
6 |
2
|
eldifad |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
7 |
|
mulre |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐵 ∈ ℝ ↔ ( 𝐴 · 𝐵 ) ∈ ℝ ) ) |
8 |
6 1 3 7
|
syl3anc |
⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ↔ ( 𝐴 · 𝐵 ) ∈ ℝ ) ) |
9 |
8
|
bicomd |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) ∈ ℝ ↔ 𝐵 ∈ ℝ ) ) |
10 |
9
|
notbid |
⊢ ( 𝜑 → ( ¬ ( 𝐴 · 𝐵 ) ∈ ℝ ↔ ¬ 𝐵 ∈ ℝ ) ) |
11 |
5 10
|
syl5bb |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) ∉ ℝ ↔ ¬ 𝐵 ∈ ℝ ) ) |
12 |
4 11
|
mpbird |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∉ ℝ ) |