Description: The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004) (Proof shortened by Andrew Salmon, 19-Nov-2011)
Ref | Expression | ||
---|---|---|---|
Assertion | mulne0b | |- ( ( A e. CC /\ B e. CC ) -> ( ( A =/= 0 /\ B =/= 0 ) <-> ( A x. B ) =/= 0 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neanior | |- ( ( A =/= 0 /\ B =/= 0 ) <-> -. ( A = 0 \/ B = 0 ) ) |
|
2 | mul0or | |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) ) ) |
|
3 | 2 | necon3abid | |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) =/= 0 <-> -. ( A = 0 \/ B = 0 ) ) ) |
4 | 1 3 | bitr4id | |- ( ( A e. CC /\ B e. CC ) -> ( ( A =/= 0 /\ B =/= 0 ) <-> ( A x. B ) =/= 0 ) ) |