Metamath Proof Explorer

Theorem mulcan2

Description: Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion mulcan2 
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A x. C ) = ( B x. C ) <-> A = B ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> A e. CC )
2 simp2 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> B e. CC )
3 simp3l 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> C e. CC )
4 simp3r 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> C =/= 0 )
5 1 2 3 4 mulcan2d 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A x. C ) = ( B x. C ) <-> A = B ) )