Theorem absltd

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

	Ref	Expression
Hypotheses	absltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	absltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	absltd	⊢ ( 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		abslt	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ 𝐴 ) < 𝐵 ↔ ( - 𝐵 < 𝐴 ∧ 𝐴 < 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) < 𝐵 ↔ ( - 𝐵 < 𝐴 ∧ 𝐴 < 𝐵 ) ) )