Metamath Proof Explorer


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 ) ) )