Description: Division into zero is zero. (Contributed by NM, 14-Mar-2005) (Proof shortened by SN, 9-Jul-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | div0 | |- ( ( A e. CC /\ A =/= 0 ) -> ( 0 / A ) = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0cn | |- 0 e. CC |
|
| 2 | eqid | |- 0 = 0 |
|
| 3 | diveq0 | |- ( ( 0 e. CC /\ A e. CC /\ A =/= 0 ) -> ( ( 0 / A ) = 0 <-> 0 = 0 ) ) |
|
| 4 | 2 3 | mpbiri | |- ( ( 0 e. CC /\ A e. CC /\ A =/= 0 ) -> ( 0 / A ) = 0 ) |
| 5 | 1 4 | mp3an1 | |- ( ( A e. CC /\ A =/= 0 ) -> ( 0 / A ) = 0 ) |