Description: Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999)
Ref | Expression | ||
---|---|---|---|
Hypotheses | divclz.1 | |- A e. CC |
|
divclz.2 | |- B e. CC |
||
Assertion | rec11i | |- ( ( A =/= 0 /\ B =/= 0 ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divclz.1 | |- A e. CC |
|
2 | divclz.2 | |- B e. CC |
|
3 | rec11 | |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) ) |
|
4 | 3 | an4s | |- ( ( ( A e. CC /\ B e. CC ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) ) |
5 | 1 2 4 | mpanl12 | |- ( ( A =/= 0 /\ B =/= 0 ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) ) |