Theorem absled

Description: Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	absltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	absltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	absled	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( - 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		absltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		absltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		absle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( - 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) ≤ 𝐵 ↔ ( - 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )