Metamath Proof Explorer
Description: Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
resqrcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
resqrcld.2 |
⊢ ( 𝜑 → 0 ≤ 𝐴 ) |
|
|
sqr11d.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
|
sqr11d.4 |
⊢ ( 𝜑 → 0 ≤ 𝐵 ) |
|
Assertion |
sqrtltd |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( √ ‘ 𝐴 ) < ( √ ‘ 𝐵 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
resqrcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
resqrcld.2 |
⊢ ( 𝜑 → 0 ≤ 𝐴 ) |
3 |
|
sqr11d.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
4 |
|
sqr11d.4 |
⊢ ( 𝜑 → 0 ≤ 𝐵 ) |
5 |
|
sqrtlt |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( √ ‘ 𝐴 ) < ( √ ‘ 𝐵 ) ) ) |
6 |
1 2 3 4 5
|
syl22anc |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( √ ‘ 𝐴 ) < ( √ ‘ 𝐵 ) ) ) |