Theorem abssuble0d

Description: Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	absltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	absltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	abssubge0d.2	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
Assertion	abssuble0d	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( 𝐵 − 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		absltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		absltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		abssubge0d.2	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
4		abssuble0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( 𝐵 − 𝐴 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( 𝐵 − 𝐴 ) )