Metamath Proof Explorer
Description: The square root of a positive real is a real. (Contributed by Mario
Carneiro, 6-Sep-2013)
|
|
Ref |
Expression |
|
Hypotheses |
sqrtthi.1 |
⊢ 𝐴 ∈ ℝ |
|
|
sqrpclii.2 |
⊢ 0 < 𝐴 |
|
Assertion |
sqrtpclii |
⊢ ( √ ‘ 𝐴 ) ∈ ℝ |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
sqrtthi.1 |
⊢ 𝐴 ∈ ℝ |
2 |
|
sqrpclii.2 |
⊢ 0 < 𝐴 |
3 |
|
0re |
⊢ 0 ∈ ℝ |
4 |
3 1 2
|
ltleii |
⊢ 0 ≤ 𝐴 |
5 |
1
|
sqrtcli |
⊢ ( 0 ≤ 𝐴 → ( √ ‘ 𝐴 ) ∈ ℝ ) |
6 |
4 5
|
ax-mp |
⊢ ( √ ‘ 𝐴 ) ∈ ℝ |