Theorem divne0

Description: The ratio of nonzero numbers is nonzero. (Contributed by NM, 28-Dec-2007)

	Ref	Expression
Assertion	divne0	|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( A / B ) =/= 0 )

Proof

Step	Hyp	Ref	Expression
1		divne0b	|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A =/= 0 <-> ( A / B ) =/= 0 ) )
2	1	3expb	|- ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( A =/= 0 <-> ( A / B ) =/= 0 ) )
3	2	biimpa	|- ( ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) ) /\ A =/= 0 ) -> ( A / B ) =/= 0 )
4	3	an32s	|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( A / B ) =/= 0 )