Step |
Hyp |
Ref |
Expression |
1 |
|
sqeq0 |
⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 ↑ 2 ) = 0 ↔ 𝐴 = 0 ) ) |
2 |
1
|
biimpa |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐴 ↑ 2 ) = 0 ) → 𝐴 = 0 ) |
3 |
2
|
3ad2antr1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ( 𝐴 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝐴 ) ∧ ( i · 𝐴 ) ∉ ℝ+ ) ) → 𝐴 = 0 ) |
4 |
|
0re |
⊢ 0 ∈ ℝ |
5 |
|
eleq1 |
⊢ ( 𝐴 = 0 → ( 𝐴 ∈ ℝ ↔ 0 ∈ ℝ ) ) |
6 |
4 5
|
mpbiri |
⊢ ( 𝐴 = 0 → 𝐴 ∈ ℝ ) |
7 |
6
|
recnd |
⊢ ( 𝐴 = 0 → 𝐴 ∈ ℂ ) |
8 |
|
sq0i |
⊢ ( 𝐴 = 0 → ( 𝐴 ↑ 2 ) = 0 ) |
9 |
|
0le0 |
⊢ 0 ≤ 0 |
10 |
|
fveq2 |
⊢ ( 𝐴 = 0 → ( ℜ ‘ 𝐴 ) = ( ℜ ‘ 0 ) ) |
11 |
|
re0 |
⊢ ( ℜ ‘ 0 ) = 0 |
12 |
10 11
|
eqtrdi |
⊢ ( 𝐴 = 0 → ( ℜ ‘ 𝐴 ) = 0 ) |
13 |
9 12
|
breqtrrid |
⊢ ( 𝐴 = 0 → 0 ≤ ( ℜ ‘ 𝐴 ) ) |
14 |
|
rennim |
⊢ ( 𝐴 ∈ ℝ → ( i · 𝐴 ) ∉ ℝ+ ) |
15 |
6 14
|
syl |
⊢ ( 𝐴 = 0 → ( i · 𝐴 ) ∉ ℝ+ ) |
16 |
8 13 15
|
3jca |
⊢ ( 𝐴 = 0 → ( ( 𝐴 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝐴 ) ∧ ( i · 𝐴 ) ∉ ℝ+ ) ) |
17 |
7 16
|
jca |
⊢ ( 𝐴 = 0 → ( 𝐴 ∈ ℂ ∧ ( ( 𝐴 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝐴 ) ∧ ( i · 𝐴 ) ∉ ℝ+ ) ) ) |
18 |
3 17
|
impbii |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ( 𝐴 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝐴 ) ∧ ( i · 𝐴 ) ∉ ℝ+ ) ) ↔ 𝐴 = 0 ) |