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Metamath Proof Explorer

Theorem lemuldiv2

Description: 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006)

Ref Expression
Assertion lemuldiv2 
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. A ) <_ B <-> A <_ ( B / C ) ) )

Proof

Step Hyp Ref Expression
1 recn 
 |-  ( A e. RR -> A e. CC )
2 recn 
 |-  ( C e. RR -> C e. CC )
3 mulcom 
 |-  ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
4 1 2 3 syl2an 
 |-  ( ( A e. RR /\ C e. RR ) -> ( A x. C ) = ( C x. A ) )
5 4 adantrr 
 |-  ( ( A e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A x. C ) = ( C x. A ) )
6 5 3adant2 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A x. C ) = ( C x. A ) )
7 6 breq1d 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) <_ B <-> ( C x. A ) <_ B ) )
8 lemuldiv 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) <_ B <-> A <_ ( B / C ) ) )
9 7 8 bitr3d 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. A ) <_ B <-> A <_ ( B / C ) ) )