| Step |
Hyp |
Ref |
Expression |
| 1 |
|
0cn |
⊢ 0 ∈ ℂ |
| 2 |
|
sqrtval |
⊢ ( 0 ∈ ℂ → ( √ ‘ 0 ) = ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) ) |
| 3 |
1 2
|
ax-mp |
⊢ ( √ ‘ 0 ) = ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) |
| 4 |
|
id |
⊢ ( 0 ∈ ℂ → 0 ∈ ℂ ) |
| 5 |
|
sqeq0 |
⊢ ( 𝑥 ∈ ℂ → ( ( 𝑥 ↑ 2 ) = 0 ↔ 𝑥 = 0 ) ) |
| 6 |
5
|
biimpa |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( 𝑥 ↑ 2 ) = 0 ) → 𝑥 = 0 ) |
| 7 |
6
|
3ad2antr1 |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) → 𝑥 = 0 ) |
| 8 |
7
|
ex |
⊢ ( 𝑥 ∈ ℂ → ( ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) → 𝑥 = 0 ) ) |
| 9 |
|
sq0i |
⊢ ( 𝑥 = 0 → ( 𝑥 ↑ 2 ) = 0 ) |
| 10 |
|
0le0 |
⊢ 0 ≤ 0 |
| 11 |
|
fveq2 |
⊢ ( 𝑥 = 0 → ( ℜ ‘ 𝑥 ) = ( ℜ ‘ 0 ) ) |
| 12 |
|
re0 |
⊢ ( ℜ ‘ 0 ) = 0 |
| 13 |
11 12
|
eqtrdi |
⊢ ( 𝑥 = 0 → ( ℜ ‘ 𝑥 ) = 0 ) |
| 14 |
10 13
|
breqtrrid |
⊢ ( 𝑥 = 0 → 0 ≤ ( ℜ ‘ 𝑥 ) ) |
| 15 |
|
0re |
⊢ 0 ∈ ℝ |
| 16 |
|
eleq1 |
⊢ ( 𝑥 = 0 → ( 𝑥 ∈ ℝ ↔ 0 ∈ ℝ ) ) |
| 17 |
15 16
|
mpbiri |
⊢ ( 𝑥 = 0 → 𝑥 ∈ ℝ ) |
| 18 |
|
rennim |
⊢ ( 𝑥 ∈ ℝ → ( i · 𝑥 ) ∉ ℝ+ ) |
| 19 |
17 18
|
syl |
⊢ ( 𝑥 = 0 → ( i · 𝑥 ) ∉ ℝ+ ) |
| 20 |
9 14 19
|
3jca |
⊢ ( 𝑥 = 0 → ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) |
| 21 |
8 20
|
impbid1 |
⊢ ( 𝑥 ∈ ℂ → ( ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ↔ 𝑥 = 0 ) ) |
| 22 |
21
|
adantl |
⊢ ( ( 0 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ↔ 𝑥 = 0 ) ) |
| 23 |
4 22
|
riota5 |
⊢ ( 0 ∈ ℂ → ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = 0 ) |
| 24 |
1 23
|
ax-mp |
⊢ ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = 0 |
| 25 |
3 24
|
eqtri |
⊢ ( √ ‘ 0 ) = 0 |