Theorem rec11ii

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
	divneq0.3	|- A =/= 0
	divneq0.4	|- B =/= 0
Assertion	rec11ii	|- ( ( 1 / A ) = ( 1 / B ) <-> A = B )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	|- A e. CC
2		divclz.2	|- B e. CC
3		divneq0.3	|- A =/= 0
4		divneq0.4	|- B =/= 0
5	1 2	rec11i	|- ( ( A =/= 0 /\ B =/= 0 ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) )
6	3 4 5	mp2an	|- ( ( 1 / A ) = ( 1 / B ) <-> A = B )