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 ) )