Step |
Hyp |
Ref |
Expression |
1 |
|
eqsqrtd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
eqsqrtd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
eqsqrtd.3 |
⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = 𝐵 ) |
4 |
|
eqsqrt2d.4 |
⊢ ( 𝜑 → 0 < ( ℜ ‘ 𝐴 ) ) |
5 |
|
0re |
⊢ 0 ∈ ℝ |
6 |
1
|
recld |
⊢ ( 𝜑 → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
7 |
|
ltle |
⊢ ( ( 0 ∈ ℝ ∧ ( ℜ ‘ 𝐴 ) ∈ ℝ ) → ( 0 < ( ℜ ‘ 𝐴 ) → 0 ≤ ( ℜ ‘ 𝐴 ) ) ) |
8 |
5 6 7
|
sylancr |
⊢ ( 𝜑 → ( 0 < ( ℜ ‘ 𝐴 ) → 0 ≤ ( ℜ ‘ 𝐴 ) ) ) |
9 |
4 8
|
mpd |
⊢ ( 𝜑 → 0 ≤ ( ℜ ‘ 𝐴 ) ) |
10 |
|
reim |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) = ( ℑ ‘ ( i · 𝐴 ) ) ) |
11 |
1 10
|
syl |
⊢ ( 𝜑 → ( ℜ ‘ 𝐴 ) = ( ℑ ‘ ( i · 𝐴 ) ) ) |
12 |
4
|
gt0ne0d |
⊢ ( 𝜑 → ( ℜ ‘ 𝐴 ) ≠ 0 ) |
13 |
11 12
|
eqnetrrd |
⊢ ( 𝜑 → ( ℑ ‘ ( i · 𝐴 ) ) ≠ 0 ) |
14 |
|
rpre |
⊢ ( ( i · 𝐴 ) ∈ ℝ+ → ( i · 𝐴 ) ∈ ℝ ) |
15 |
14
|
reim0d |
⊢ ( ( i · 𝐴 ) ∈ ℝ+ → ( ℑ ‘ ( i · 𝐴 ) ) = 0 ) |
16 |
15
|
necon3ai |
⊢ ( ( ℑ ‘ ( i · 𝐴 ) ) ≠ 0 → ¬ ( i · 𝐴 ) ∈ ℝ+ ) |
17 |
13 16
|
syl |
⊢ ( 𝜑 → ¬ ( i · 𝐴 ) ∈ ℝ+ ) |
18 |
1 2 3 9 17
|
eqsqrtd |
⊢ ( 𝜑 → 𝐴 = ( √ ‘ 𝐵 ) ) |