Metamath Proof Explorer


Theorem divmulzi

Description: Relationship between division and multiplication. (Contributed by NM, 8-May-1999) (Revised by Mario Carneiro, 17-Feb-2014)

Ref Expression
Hypotheses divclz.1
|- A e. CC
divclz.2
|- B e. CC
divmulz.3
|- C e. CC
Assertion divmulzi
|- ( B =/= 0 -> ( ( A / B ) = C <-> ( B x. C ) = A ) )

Proof

Step Hyp Ref Expression
1 divclz.1
 |-  A e. CC
2 divclz.2
 |-  B e. CC
3 divmulz.3
 |-  C e. CC
4 divmul
 |-  ( ( A e. CC /\ C e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A / B ) = C <-> ( B x. C ) = A ) )
5 1 3 4 mp3an12
 |-  ( ( B e. CC /\ B =/= 0 ) -> ( ( A / B ) = C <-> ( B x. C ) = A ) )
6 2 5 mpan
 |-  ( B =/= 0 -> ( ( A / B ) = C <-> ( B x. C ) = A ) )