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Metamath Proof Explorer

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 ) ) )