Theorem divmuli

Description: Relationship between division and multiplication. (Contributed by NM, 2-Feb-1995) (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
	divmul.4	|- B =/= 0
Assertion	divmuli	|- ( ( 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.4	|- B =/= 0
5	1 2 3	divmulzi	|- ( B =/= 0 -> ( ( A / B ) = C <-> ( B x. C ) = A ) )
6	4 5	ax-mp	|- ( ( A / B ) = C <-> ( B x. C ) = A )