Theorem div11i

Description: One-to-one relationship for division. (Contributed by NM, 20-Aug-2001)

	Ref	Expression
Hypotheses	divclz.1	|- A e. CC
	divclz.2	|- B e. CC
	divmulz.3	|- C e. CC
	divass.4	|- C =/= 0
Assertion	div11i	|- ( ( A / C ) = ( B / C ) <-> A = B )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	|- A e. CC
2		divclz.2	|- B e. CC
3		divmulz.3	|- C e. CC
4		divass.4	|- C =/= 0
5	3 4	pm3.2i	|- ( C e. CC /\ C =/= 0 )
6		div11	|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / C ) = ( B / C ) <-> A = B ) )
7	1 2 5 6	mp3an	|- ( ( A / C ) = ( B / C ) <-> A = B )