Theorem le2sqi

Description: The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999)

	Ref	Expression
Hypotheses	resqcl.1	⊢ 𝐴 ∈ ℝ
	lt2sq.2	⊢ 𝐵 ∈ ℝ
Assertion	le2sqi	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		resqcl.1	⊢ 𝐴 ∈ ℝ
2		lt2sq.2	⊢ 𝐵 ∈ ℝ
3	1 2	le2msqi	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 · 𝐴 ) ≤ ( 𝐵 · 𝐵 ) ) )
4	1	recni	⊢ 𝐴 ∈ ℂ
5	4	sqvali	⊢ ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 )
6	2	recni	⊢ 𝐵 ∈ ℂ
7	6	sqvali	⊢ ( 𝐵 ↑ 2 ) = ( 𝐵 · 𝐵 )
8	5 7	breq12i	⊢ ( ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ↔ ( 𝐴 · 𝐴 ) ≤ ( 𝐵 · 𝐵 ) )
9	3 8	bitr4di	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )