Metamath Proof Explorer

Theorem reccld

Description: Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1 
|- ( ph -> A e. CC )
reccld.2 
|- ( ph -> A =/= 0 )
Assertion reccld 
|- ( ph -> ( 1 / A ) e. CC )

Proof

Step Hyp Ref Expression
1 div1d.1 
 |-  ( ph -> A e. CC )
2 reccld.2 
 |-  ( ph -> A =/= 0 )
3 reccl 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) e. CC )
4 1 2 3 syl2anc 
 |-  ( ph -> ( 1 / A ) e. CC )