Metamath Proof Explorer

Theorem xlemul2

Description: Extended real version of lemul2 . (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xlemul2 
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A <_ B <-> ( C *e A ) <_ ( C *e B ) ) )

Proof

Step Hyp Ref Expression
1 xlemul1 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A <_ B <-> ( A *e C ) <_ ( B *e C ) ) )
2 simp1 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> A e. RR* )
3 rpxr 
 |-  ( C e. RR+ -> C e. RR* )
4 3 3ad2ant3 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> C e. RR* )
5 xmulcom 
 |-  ( ( A e. RR* /\ C e. RR* ) -> ( A *e C ) = ( C *e A ) )
6 2 4 5 syl2anc 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A *e C ) = ( C *e A ) )
7 simp2 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> B e. RR* )
8 xmulcom 
 |-  ( ( B e. RR* /\ C e. RR* ) -> ( B *e C ) = ( C *e B ) )
9 7 4 8 syl2anc 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( B *e C ) = ( C *e B ) )
10 6 9 breq12d 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( A *e C ) <_ ( B *e C ) <-> ( C *e A ) <_ ( C *e B ) ) )
11 1 10 bitrd 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A <_ B <-> ( C *e A ) <_ ( C *e B ) ) )