Theorem lt2sq

Description: The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006)

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

Proof

Step	Hyp	Ref	Expression
1		lt2msq	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A < B <-> ( A x. A ) < ( B x. B ) ) )
2		recn	|- ( A e. RR -> A e. CC )
3		recn	|- ( B e. RR -> B e. CC )
4		sqval	|- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
5		sqval	|- ( B e. CC -> ( B ^ 2 ) = ( B x. B ) )
6	4 5	breqan12d	|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) < ( B ^ 2 ) <-> ( A x. A ) < ( B x. B ) ) )
7	2 3 6	syl2an	|- ( ( A e. RR /\ B e. RR ) -> ( ( A ^ 2 ) < ( B ^ 2 ) <-> ( A x. A ) < ( B x. B ) ) )
8	7	ad2ant2r	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A ^ 2 ) < ( B ^ 2 ) <-> ( A x. A ) < ( B x. B ) ) )
9	1 8	bitr4d	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A < B <-> ( A ^ 2 ) < ( B ^ 2 ) ) )