Theorem absdifled

Description: The absolute value of a difference 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 )
	absltd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
Assertion	absdifled	⊢ ( 𝜑 → ( ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ 𝐶 ↔ ( ( 𝐵 − 𝐶 ) ≤ 𝐴 ∧ 𝐴 ≤ ( 𝐵 + 𝐶 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		absltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		absltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		absltd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		absdifle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ 𝐶 ↔ ( ( 𝐵 − 𝐶 ) ≤ 𝐴 ∧ 𝐴 ≤ ( 𝐵 + 𝐶 ) ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ 𝐶 ↔ ( ( 𝐵 − 𝐶 ) ≤ 𝐴 ∧ 𝐴 ≤ ( 𝐵 + 𝐶 ) ) ) )