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Metamath Proof Explorer

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 ) ) ) )