Theorem abs2difabsi

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

	Ref	Expression
Hypotheses	abs2difabsi.1	|- A e. CC
	abs2difabsi.2	|- B e. CC
Assertion	abs2difabsi	|- ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) )

Proof

Step	Hyp	Ref	Expression
1		abs2difabsi.1	|- A e. CC
2		abs2difabsi.2	|- B e. CC
3		abs2difabs	|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) )
4	1 2 3	mp2an	|- ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) )