Step |
Hyp |
Ref |
Expression |
1 |
|
eqsqrtd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
eqsqrtd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
eqsqrtd.3 |
⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = 𝐵 ) |
4 |
|
eqsqrtd.4 |
⊢ ( 𝜑 → 0 ≤ ( ℜ ‘ 𝐴 ) ) |
5 |
|
eqsqrtd.5 |
⊢ ( 𝜑 → ¬ ( i · 𝐴 ) ∈ ℝ+ ) |
6 |
|
sqreu |
⊢ ( 𝐵 ∈ ℂ → ∃! 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) |
7 |
|
reurmo |
⊢ ( ∃! 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) → ∃* 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) |
8 |
2 6 7
|
3syl |
⊢ ( 𝜑 → ∃* 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) |
9 |
|
df-nel |
⊢ ( ( i · 𝐴 ) ∉ ℝ+ ↔ ¬ ( i · 𝐴 ) ∈ ℝ+ ) |
10 |
5 9
|
sylibr |
⊢ ( 𝜑 → ( i · 𝐴 ) ∉ ℝ+ ) |
11 |
3 4 10
|
3jca |
⊢ ( 𝜑 → ( ( 𝐴 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝐴 ) ∧ ( i · 𝐴 ) ∉ ℝ+ ) ) |
12 |
|
sqrtcl |
⊢ ( 𝐵 ∈ ℂ → ( √ ‘ 𝐵 ) ∈ ℂ ) |
13 |
2 12
|
syl |
⊢ ( 𝜑 → ( √ ‘ 𝐵 ) ∈ ℂ ) |
14 |
|
sqrtthlem |
⊢ ( 𝐵 ∈ ℂ → ( ( ( √ ‘ 𝐵 ) ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐵 ) ) ∧ ( i · ( √ ‘ 𝐵 ) ) ∉ ℝ+ ) ) |
15 |
2 14
|
syl |
⊢ ( 𝜑 → ( ( ( √ ‘ 𝐵 ) ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐵 ) ) ∧ ( i · ( √ ‘ 𝐵 ) ) ∉ ℝ+ ) ) |
16 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ↑ 2 ) = ( 𝐴 ↑ 2 ) ) |
17 |
16
|
eqeq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ↑ 2 ) = 𝐵 ↔ ( 𝐴 ↑ 2 ) = 𝐵 ) ) |
18 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( ℜ ‘ 𝑥 ) = ( ℜ ‘ 𝐴 ) ) |
19 |
18
|
breq2d |
⊢ ( 𝑥 = 𝐴 → ( 0 ≤ ( ℜ ‘ 𝑥 ) ↔ 0 ≤ ( ℜ ‘ 𝐴 ) ) ) |
20 |
|
oveq2 |
⊢ ( 𝑥 = 𝐴 → ( i · 𝑥 ) = ( i · 𝐴 ) ) |
21 |
|
neleq1 |
⊢ ( ( i · 𝑥 ) = ( i · 𝐴 ) → ( ( i · 𝑥 ) ∉ ℝ+ ↔ ( i · 𝐴 ) ∉ ℝ+ ) ) |
22 |
20 21
|
syl |
⊢ ( 𝑥 = 𝐴 → ( ( i · 𝑥 ) ∉ ℝ+ ↔ ( i · 𝐴 ) ∉ ℝ+ ) ) |
23 |
17 19 22
|
3anbi123d |
⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ↔ ( ( 𝐴 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝐴 ) ∧ ( i · 𝐴 ) ∉ ℝ+ ) ) ) |
24 |
|
oveq1 |
⊢ ( 𝑥 = ( √ ‘ 𝐵 ) → ( 𝑥 ↑ 2 ) = ( ( √ ‘ 𝐵 ) ↑ 2 ) ) |
25 |
24
|
eqeq1d |
⊢ ( 𝑥 = ( √ ‘ 𝐵 ) → ( ( 𝑥 ↑ 2 ) = 𝐵 ↔ ( ( √ ‘ 𝐵 ) ↑ 2 ) = 𝐵 ) ) |
26 |
|
fveq2 |
⊢ ( 𝑥 = ( √ ‘ 𝐵 ) → ( ℜ ‘ 𝑥 ) = ( ℜ ‘ ( √ ‘ 𝐵 ) ) ) |
27 |
26
|
breq2d |
⊢ ( 𝑥 = ( √ ‘ 𝐵 ) → ( 0 ≤ ( ℜ ‘ 𝑥 ) ↔ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐵 ) ) ) ) |
28 |
|
oveq2 |
⊢ ( 𝑥 = ( √ ‘ 𝐵 ) → ( i · 𝑥 ) = ( i · ( √ ‘ 𝐵 ) ) ) |
29 |
|
neleq1 |
⊢ ( ( i · 𝑥 ) = ( i · ( √ ‘ 𝐵 ) ) → ( ( i · 𝑥 ) ∉ ℝ+ ↔ ( i · ( √ ‘ 𝐵 ) ) ∉ ℝ+ ) ) |
30 |
28 29
|
syl |
⊢ ( 𝑥 = ( √ ‘ 𝐵 ) → ( ( i · 𝑥 ) ∉ ℝ+ ↔ ( i · ( √ ‘ 𝐵 ) ) ∉ ℝ+ ) ) |
31 |
25 27 30
|
3anbi123d |
⊢ ( 𝑥 = ( √ ‘ 𝐵 ) → ( ( ( 𝑥 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ↔ ( ( ( √ ‘ 𝐵 ) ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐵 ) ) ∧ ( i · ( √ ‘ 𝐵 ) ) ∉ ℝ+ ) ) ) |
32 |
23 31
|
rmoi |
⊢ ( ( ∃* 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ∧ ( 𝐴 ∈ ℂ ∧ ( ( 𝐴 ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ 𝐴 ) ∧ ( i · 𝐴 ) ∉ ℝ+ ) ) ∧ ( ( √ ‘ 𝐵 ) ∈ ℂ ∧ ( ( ( √ ‘ 𝐵 ) ↑ 2 ) = 𝐵 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐵 ) ) ∧ ( i · ( √ ‘ 𝐵 ) ) ∉ ℝ+ ) ) ) → 𝐴 = ( √ ‘ 𝐵 ) ) |
33 |
8 1 11 13 15 32
|
syl122anc |
⊢ ( 𝜑 → 𝐴 = ( √ ‘ 𝐵 ) ) |