Metamath Proof Explorer

Theorem diveq0

Description: A ratio is zero iff the numerator is zero. (Contributed by NM, 20-Apr-2006) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion diveq0 
|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 0 <-> A = 0 ) )

Proof

Step Hyp Ref Expression
1 0cn 
 |-  0 e. CC
2 divmul2 
 |-  ( ( A e. CC /\ 0 e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A / B ) = 0 <-> A = ( B x. 0 ) ) )
3 1 2 mp3an2 
 |-  ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A / B ) = 0 <-> A = ( B x. 0 ) ) )
4 3 3impb 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 0 <-> A = ( B x. 0 ) ) )
5 simp2 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> B e. CC )
6 5 mul01d 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( B x. 0 ) = 0 )
7 6 eqeq2d 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A = ( B x. 0 ) <-> A = 0 ) )
8 4 7 bitrd 
 |-  ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 0 <-> A = 0 ) )