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Metamath Proof Explorer

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 ) )