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Metamath Proof Explorer

Theorem xltmul2

Description: Extended real version of ltmul2 . (Contributed by Mario Carneiro, 8-Sep-2015)

Ref Expression
Assertion xltmul2 
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A < B <-> ( C *e A ) < ( C *e B ) ) )

Proof

Step Hyp Ref Expression
1 xltmul1 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A < B <-> ( A *e C ) < ( B *e C ) ) )
2 rpxr 
 |-  ( C e. RR+ -> C e. RR* )
3 xmulcom 
 |-  ( ( A e. RR* /\ C e. RR* ) -> ( A *e C ) = ( C *e A ) )
4 3 3adant2 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A *e C ) = ( C *e A ) )
5 xmulcom 
 |-  ( ( B e. RR* /\ C e. RR* ) -> ( B *e C ) = ( C *e B ) )
6 5 3adant1 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B *e C ) = ( C *e B ) )
7 4 6 breq12d 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A *e C ) < ( B *e C ) <-> ( C *e A ) < ( C *e B ) ) )
8 2 7 syl3an3 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( A *e C ) < ( B *e C ) <-> ( C *e A ) < ( C *e B ) ) )
9 1 8 bitrd 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A < B <-> ( C *e A ) < ( C *e B ) ) )