Metamath Proof Explorer


Theorem rec11i

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 ) )

Proof

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 ) )