Theorem le2sqd

Description: The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	resqcld.1	|- ( ph -> A e. RR )
	lt2sqd.2	|- ( ph -> B e. RR )
	lt2sqd.3	|- ( ph -> 0 <_ A )
	lt2sqd.4	|- ( ph -> 0 <_ B )
Assertion	le2sqd	|- ( ph -> ( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		resqcld.1	|- ( ph -> A e. RR )
2		lt2sqd.2	|- ( ph -> B e. RR )
3		lt2sqd.3	|- ( ph -> 0 <_ A )
4		lt2sqd.4	|- ( ph -> 0 <_ B )
5		le2sq	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) )
6	1 3 2 4 5	syl22anc	|- ( ph -> ( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) )