Theorem abssuble0

Description: Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008)

	Ref	Expression
Assertion	abssuble0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( 𝐵 − 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2		recn	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
3		abssub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )
5	4	3adant3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )
6		abssubge0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( abs ‘ ( 𝐵 − 𝐴 ) ) = ( 𝐵 − 𝐴 ) )
7	5 6	eqtrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( 𝐵 − 𝐴 ) )