Theorem lenegsq

Description: Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006)

	Ref	Expression
Assertion	lenegsq	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( ( 𝐴 ≤ 𝐵 ∧ - 𝐴 ≤ 𝐵 ) ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2		abscl	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ )
3		absge0	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( abs ‘ 𝐴 ) )
4	2 3	jca	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝐴 ) ) )
5	1 4	syl	⊢ ( 𝐴 ∈ ℝ → ( ( abs ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝐴 ) ) )
6		le2sq	⊢ ( ( ( ( abs ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝐴 ) ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )
7	5 6	sylan	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )
8		absle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( - 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )
9		lenegcon1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( - 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ 𝐴 ) )
10	9	anbi1d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( - 𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐵 ) ↔ ( - 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )
11		ancom	⊢ ( ( - 𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐵 ) ↔ ( 𝐴 ≤ 𝐵 ∧ - 𝐴 ≤ 𝐵 ) )
12	10 11	bitr3di	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( - 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ↔ ( 𝐴 ≤ 𝐵 ∧ - 𝐴 ≤ 𝐵 ) ) )
13	8 12	bitrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ - 𝐴 ≤ 𝐵 ) ) )
14	13	adantrr	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ - 𝐴 ≤ 𝐵 ) ) )
15		absresq	⊢ ( 𝐴 ∈ ℝ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ↑ 2 ) )
16	15	breq1d	⊢ ( 𝐴 ∈ ℝ → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )
17	16	adantr	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )
18	7 14 17	3bitr3d	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ≤ 𝐵 ∧ - 𝐴 ≤ 𝐵 ) ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )
19	18	3impb	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( ( 𝐴 ≤ 𝐵 ∧ - 𝐴 ≤ 𝐵 ) ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )