Metamath Proof Explorer

Theorem xltmul1

Description: Extended real version of ltmul1 . (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xltmul1 
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A < B <-> ( A *e C ) < ( B *e C ) ) )

Proof

Step Hyp Ref Expression
1 xlemul1 
 |-  ( ( B e. RR* /\ A e. RR* /\ C e. RR+ ) -> ( B <_ A <-> ( B *e C ) <_ ( A *e C ) ) )
2 1 3com12 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( B <_ A <-> ( B *e C ) <_ ( A *e C ) ) )
3 2 notbid 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( -. B <_ A <-> -. ( B *e C ) <_ ( A *e C ) ) )
4 xrltnle 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -. B <_ A ) )
5 4 3adant3 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A < B <-> -. B <_ A ) )
6 simp1 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> A e. RR* )
7 rpxr 
 |-  ( C e. RR+ -> C e. RR* )
8 7 3ad2ant3 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> C e. RR* )
9 xmulcl 
 |-  ( ( A e. RR* /\ C e. RR* ) -> ( A *e C ) e. RR* )
10 6 8 9 syl2anc 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A *e C ) e. RR* )
11 simp2 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> B e. RR* )
12 xmulcl 
 |-  ( ( B e. RR* /\ C e. RR* ) -> ( B *e C ) e. RR* )
13 11 8 12 syl2anc 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( B *e C ) e. RR* )
14 xrltnle 
 |-  ( ( ( A *e C ) e. RR* /\ ( B *e C ) e. RR* ) -> ( ( A *e C ) < ( B *e C ) <-> -. ( B *e C ) <_ ( A *e C ) ) )
15 10 13 14 syl2anc 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( A *e C ) < ( B *e C ) <-> -. ( B *e C ) <_ ( A *e C ) ) )
16 3 5 15 3bitr4d 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A < B <-> ( A *e C ) < ( B *e C ) ) )