Theorem le2sqi

Description: The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999)

	Ref	Expression
Hypotheses	resqcl.1	|- A e. RR
	lt2sq.2	|- B e. RR
Assertion	le2sqi	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		resqcl.1	|- A e. RR
2		lt2sq.2	|- B e. RR
3	1 2	le2msqi	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) )
4	1	recni	|- A e. CC
5	4	sqvali	|- ( A ^ 2 ) = ( A x. A )
6	2	recni	|- B e. CC
7	6	sqvali	|- ( B ^ 2 ) = ( B x. B )
8	5 7	breq12i	|- ( ( A ^ 2 ) <_ ( B ^ 2 ) <-> ( A x. A ) <_ ( B x. B ) )
9	3 8	bitr4di	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) )