Metamath Proof Explorer

Theorem recne0

Description: The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion recne0 
|- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) =/= 0 )

Proof

Step Hyp Ref Expression
1 ax-1cn 
 |-  1 e. CC
2 ax-1ne0 
 |-  1 =/= 0
3 divne0 
 |-  ( ( ( 1 e. CC /\ 1 =/= 0 ) /\ ( A e. CC /\ A =/= 0 ) ) -> ( 1 / A ) =/= 0 )
4 1 2 3 mpanl12 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) =/= 0 )