Theorem abssubd

Description: Swapping order of subtraction doesn't change the absolute value. Example of Apostol p. 363. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	abssubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	abssubd	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		abssubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		abssub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )