Metamath Proof Explorer

Theorem recreci

Description: A number is equal to the reciprocal of its reciprocal. Theorem I.10 of Apostol p. 18. (Contributed by NM, 9-Feb-1995)

Ref Expression
Hypotheses divclz.1 
|- A e. CC
reccl.2 
|- A =/= 0
Assertion recreci 
|- ( 1 / ( 1 / A ) ) = A

Proof

Step Hyp Ref Expression
1 divclz.1 
 |-  A e. CC
2 reccl.2 
 |-  A =/= 0
3 recrec 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( 1 / ( 1 / A ) ) = A )
4 1 2 3 mp2an 
 |-  ( 1 / ( 1 / A ) ) = A