Theorem abslei

Description: Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005)

	Ref	Expression
Hypotheses	sqrtthi.1	⊢ 𝐴 ∈ ℝ
	sqr11.1	⊢ 𝐵 ∈ ℝ
Assertion	abslei	⊢ ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( - 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	⊢ 𝐴 ∈ ℝ
2		sqr11.1	⊢ 𝐵 ∈ ℝ
3		absle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( - 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( - 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) )