Theorem abssubi

Description: Swapping order of subtraction doesn't change the absolute value. Example of Apostol p. 363. (Contributed by NM, 1-Oct-1999)

	Ref	Expression
Hypotheses	absvalsqi.1	⊢ 𝐴 ∈ ℂ
	abssub.2	⊢ 𝐵 ∈ ℂ
Assertion	abssubi	⊢ ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	⊢ 𝐴 ∈ ℂ
2		abssub.2	⊢ 𝐵 ∈ ℂ
3		abssub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )
4	1 2 3	mp2an	⊢ ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) )