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Metamath Proof Explorer

Theorem le2sq

Description: The square function is nondecreasing on nonnegative reals. (Contributed by NM, 18-Oct-1999)

Ref Expression
Assertion le2sq 
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) )

Proof

Step Hyp Ref Expression
1 le2msq 
 |-  ( ( ( 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 ) ) )