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 ) )