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Metamath Proof Explorer

Theorem lemuldiv

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

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

Proof

Step Hyp Ref Expression
1 ltdivmul2 
 |-  ( ( B e. RR /\ A e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( B / C ) < A <-> B < ( A x. C ) ) )
2 1 3com12 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( B / C ) < A <-> B < ( A x. C ) ) )
3 2 notbid 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( -. ( B / C ) < A <-> -. B < ( A x. C ) ) )
4 simp1 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
5 gt0ne0 
 |-  ( ( C e. RR /\ 0 < C ) -> C =/= 0 )
6 5 3adant1 
 |-  ( ( B e. RR /\ C e. RR /\ 0 < C ) -> C =/= 0 )
7 redivcl 
 |-  ( ( B e. RR /\ C e. RR /\ C =/= 0 ) -> ( B / C ) e. RR )
8 6 7 syld3an3 
 |-  ( ( B e. RR /\ C e. RR /\ 0 < C ) -> ( B / C ) e. RR )
9 8 3expb 
 |-  ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B / C ) e. RR )
10 9 3adant1 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B / C ) e. RR )
11 4 10 lenltd 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ ( B / C ) <-> -. ( B / C ) < A ) )
12 remulcl 
 |-  ( ( A e. RR /\ C e. RR ) -> ( A x. C ) e. RR )
13 12 3adant2 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A x. C ) e. RR )
14 simp2 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
15 13 14 lenltd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) <_ B <-> -. B < ( A x. C ) ) )
16 15 3adant3r 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) <_ B <-> -. B < ( A x. C ) ) )
17 3 11 16 3bitr4rd 
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) <_ B <-> A <_ ( B / C ) ) )