Theorem abs2sqlei

Description: The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007)

	Ref	Expression
Hypotheses	abs2sqlei.1	|- A e. CC
	abs2sqlei.2	|- B e. CC
Assertion	abs2sqlei	|- ( ( abs ` A ) <_ ( abs ` B ) <-> ( ( abs ` A ) ^ 2 ) <_ ( ( abs ` B ) ^ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		abs2sqlei.1	|- A e. CC
2		abs2sqlei.2	|- B e. CC
3	1	absge0i	|- 0 <_ ( abs ` A )
4	2	absge0i	|- 0 <_ ( abs ` B )
5	1	abscli	|- ( abs ` A ) e. RR
6	2	abscli	|- ( abs ` B ) e. RR
7	5 6	le2sqi	|- ( ( 0 <_ ( abs ` A ) /\ 0 <_ ( abs ` B ) ) -> ( ( abs ` A ) <_ ( abs ` B ) <-> ( ( abs ` A ) ^ 2 ) <_ ( ( abs ` B ) ^ 2 ) ) )
8	3 4 7	mp2an	|- ( ( abs ` A ) <_ ( abs ` B ) <-> ( ( abs ` A ) ^ 2 ) <_ ( ( abs ` B ) ^ 2 ) )