Metamath Proof Explorer
Description: The square root of a nonnegative real is a real. (Contributed by Mario
Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
resqrcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
resqrcld.2 |
⊢ ( 𝜑 → 0 ≤ 𝐴 ) |
|
Assertion |
resqrtcld |
⊢ ( 𝜑 → ( √ ‘ 𝐴 ) ∈ ℝ ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
resqrcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
resqrcld.2 |
⊢ ( 𝜑 → 0 ≤ 𝐴 ) |
3 |
|
resqrtcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( √ ‘ 𝐴 ) ∈ ℝ ) |