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	|- ( ph -> A e. CC )
	abssubd.2	|- ( ph -> B e. CC )
Assertion	abssubd	|- ( ph -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	|- ( ph -> A e. CC )
2		abssubd.2	|- ( ph -> B e. CC )
3		abssub	|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )