Theorem sqrtlei

Description: Square root is monotonic. (Contributed by NM, 3-Aug-1999)

	Ref	Expression
Hypotheses	sqrtthi.1	⊢ 𝐴 ∈ ℝ
	sqr11.1	⊢ 𝐵 ∈ ℝ
Assertion	sqrtlei	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( 𝐴 ≤ 𝐵 ↔ ( √ ‘ 𝐴 ) ≤ ( √ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	⊢ 𝐴 ∈ ℝ
2		sqr11.1	⊢ 𝐵 ∈ ℝ
3		sqrtle	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( √ ‘ 𝐴 ) ≤ ( √ ‘ 𝐵 ) ) )
4	2 3	mpanr1	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 0 ≤ 𝐵 ) → ( 𝐴 ≤ 𝐵 ↔ ( √ ‘ 𝐴 ) ≤ ( √ ‘ 𝐵 ) ) )
5	1 4	mpanl1	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( 𝐴 ≤ 𝐵 ↔ ( √ ‘ 𝐴 ) ≤ ( √ ‘ 𝐵 ) ) )