Theorem sqrtled

Description: Square root is 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	sqrtled	⊢ ( 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		sqrtle	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( √ ‘ 𝐴 ) ≤ ( √ ‘ 𝐵 ) ) )
6	1 2 3 4 5	syl22anc	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( √ ‘ 𝐴 ) ≤ ( √ ‘ 𝐵 ) ) )