Step |
Hyp |
Ref |
Expression |
1 |
|
resqrtcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ ) |
2 |
|
sqrtge0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 0 ≤ ( √ ‘ 𝐴 ) ) |
3 |
1 2
|
jca |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( √ ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( √ ‘ 𝐴 ) ) ) |
4 |
|
resqrtcl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( √ ‘ 𝐵 ) ∈ ℝ ) |
5 |
|
sqrtge0 |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → 0 ≤ ( √ ‘ 𝐵 ) ) |
6 |
4 5
|
jca |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐵 ) ∈ ℝ ∧ 0 ≤ ( √ ‘ 𝐵 ) ) ) |
7 |
|
sq11 |
⊢ ( ( ( ( √ ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( √ ‘ 𝐴 ) ) ∧ ( ( √ ‘ 𝐵 ) ∈ ℝ ∧ 0 ≤ ( √ ‘ 𝐵 ) ) ) → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ 𝐵 ) ↑ 2 ) ↔ ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ) ) |
8 |
3 6 7
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ 𝐵 ) ↑ 2 ) ↔ ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ) ) |
9 |
|
resqrtth |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ) |
10 |
|
resqrtth |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐵 ) ↑ 2 ) = 𝐵 ) |
11 |
9 10
|
eqeqan12d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ 𝐵 ) ↑ 2 ) ↔ 𝐴 = 𝐵 ) ) |
12 |
8 11
|
bitr3d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |