Metamath Proof Explorer


Theorem sqrtled

Description: Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses resqrcld.1 ( 𝜑𝐴 ∈ ℝ )
resqrcld.2 ( 𝜑 → 0 ≤ 𝐴 )
sqr11d.3 ( 𝜑𝐵 ∈ ℝ )
sqr11d.4 ( 𝜑 → 0 ≤ 𝐵 )
Assertion sqrtled ( 𝜑 → ( 𝐴𝐵 ↔ ( √ ‘ 𝐴 ) ≤ ( √ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 resqrcld.1 ( 𝜑𝐴 ∈ ℝ )
2 resqrcld.2 ( 𝜑 → 0 ≤ 𝐴 )
3 sqr11d.3 ( 𝜑𝐵 ∈ ℝ )
4 sqr11d.4 ( 𝜑 → 0 ≤ 𝐵 )
5 sqrtle ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴𝐵 ↔ ( √ ‘ 𝐴 ) ≤ ( √ ‘ 𝐵 ) ) )
6 1 2 3 4 5 syl22anc ( 𝜑 → ( 𝐴𝐵 ↔ ( √ ‘ 𝐴 ) ≤ ( √ ‘ 𝐵 ) ) )