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Metamath Proof Explorer
Description: The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypothesis |
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
Assertion |
sqrtrege0d |
⊢ ( 𝜑 → 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
| 2 |
|
sqrtrege0 |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ) |
| 3 |
1 2
|
syl |
⊢ ( 𝜑 → 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ) |