Step |
Hyp |
Ref |
Expression |
1 |
|
2cnne0 |
⊢ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) |
2 |
|
0cxp |
⊢ ( ( 2 ∈ ℂ ∧ 2 ≠ 0 ) → ( 0 ↑𝑐 2 ) = 0 ) |
3 |
1 2
|
ax-mp |
⊢ ( 0 ↑𝑐 2 ) = 0 |
4 |
|
fveq2 |
⊢ ( 𝐴 = 0 → ( √ ‘ 𝐴 ) = ( √ ‘ 0 ) ) |
5 |
|
sqrt0 |
⊢ ( √ ‘ 0 ) = 0 |
6 |
4 5
|
eqtrdi |
⊢ ( 𝐴 = 0 → ( √ ‘ 𝐴 ) = 0 ) |
7 |
6
|
oveq1d |
⊢ ( 𝐴 = 0 → ( ( √ ‘ 𝐴 ) ↑𝑐 2 ) = ( 0 ↑𝑐 2 ) ) |
8 |
|
id |
⊢ ( 𝐴 = 0 → 𝐴 = 0 ) |
9 |
3 7 8
|
3eqtr4a |
⊢ ( 𝐴 = 0 → ( ( √ ‘ 𝐴 ) ↑𝑐 2 ) = 𝐴 ) |
10 |
9
|
a1d |
⊢ ( 𝐴 = 0 → ( 𝐴 ∈ ℂ → ( ( √ ‘ 𝐴 ) ↑𝑐 2 ) = 𝐴 ) ) |
11 |
|
sqrtcl |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 𝐴 ) ∈ ℂ ) |
12 |
11
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( √ ‘ 𝐴 ) ∈ ℂ ) |
13 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( √ ‘ 𝐴 ) = 0 ) → 𝐴 ∈ ℂ ) |
14 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( √ ‘ 𝐴 ) = 0 ) → ( √ ‘ 𝐴 ) = 0 ) |
15 |
13 14
|
sqr00d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( √ ‘ 𝐴 ) = 0 ) → 𝐴 = 0 ) |
16 |
15
|
ex |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ 𝐴 ) = 0 → 𝐴 = 0 ) ) |
17 |
16
|
necon3d |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ≠ 0 → ( √ ‘ 𝐴 ) ≠ 0 ) ) |
18 |
17
|
imp |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( √ ‘ 𝐴 ) ≠ 0 ) |
19 |
|
2z |
⊢ 2 ∈ ℤ |
20 |
19
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → 2 ∈ ℤ ) |
21 |
12 18 20
|
cxpexpzd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( √ ‘ 𝐴 ) ↑𝑐 2 ) = ( ( √ ‘ 𝐴 ) ↑ 2 ) ) |
22 |
|
sqrtth |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ) |
23 |
22
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ) |
24 |
21 23
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( √ ‘ 𝐴 ) ↑𝑐 2 ) = 𝐴 ) |
25 |
24
|
expcom |
⊢ ( 𝐴 ≠ 0 → ( 𝐴 ∈ ℂ → ( ( √ ‘ 𝐴 ) ↑𝑐 2 ) = 𝐴 ) ) |
26 |
10 25
|
pm2.61ine |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ 𝐴 ) ↑𝑐 2 ) = 𝐴 ) |