Theorem le2sqd

Description: The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	lt2sqd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	lt2sqd.3	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	lt2sqd.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
Assertion	le2sqd	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		lt2sqd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		lt2sqd.3	⊢ ( 𝜑 → 0 ≤ 𝐴 )
4		lt2sqd.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
5		le2sq	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )
6	1 3 2 4 5	syl22anc	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )