Theorem abs2difi

Description: Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007)

	Ref	Expression
Hypotheses	abs2difi.1	|- A e. CC
	abs2difi.2	|- B e. CC
Assertion	abs2difi	|- ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) )

Proof

Step	Hyp	Ref	Expression
1		abs2difi.1	|- A e. CC
2		abs2difi.2	|- B e. CC
3		abs2dif	|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )
4	1 2 3	mp2an	|- ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) )