Theorem mulcani

Description: Cancellation law for multiplication. Theorem I.7 of Apostol p. 18. (Contributed by NM, 26-Jan-1995)

	Ref	Expression
Hypotheses	mulcan.1	|- A e. CC
	mulcan.2	|- B e. CC
	mulcan.3	|- C e. CC
	mulcan.4	|- C =/= 0
Assertion	mulcani	|- ( ( C x. A ) = ( C x. B ) <-> A = B )

Proof

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