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Metamath Proof Explorer

Theorem ltmuldiv2

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004)

Ref Expression
Assertion ltmuldiv2 
|- ( ( 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 ltmuldiv 
 |-  ( ( 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 ) ) )