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Metamath Proof Explorer

Theorem drnginvrcl

Description: Closure of the multiplicative inverse in a division ring. ( reccl analog). (Contributed by NM, 19-Apr-2014)

Ref Expression
Hypotheses drnginvrcl.b 
|- B = ( Base ` R )
drnginvrcl.z 
|- .0. = ( 0g ` R )
drnginvrcl.i 
|- I = ( invr ` R )
Assertion drnginvrcl 
|- ( ( R e. DivRing /\ X e. B /\ X =/= .0. ) -> ( I ` X ) e. B )

Proof

Step Hyp Ref Expression
1 drnginvrcl.b 
 |-  B = ( Base ` R )
2 drnginvrcl.z 
 |-  .0. = ( 0g ` R )
3 drnginvrcl.i 
 |-  I = ( invr ` R )
4 eqid 
 |-  ( Unit ` R ) = ( Unit ` R )
5 1 4 2 drngunit 
 |-  ( R e. DivRing -> ( X e. ( Unit ` R ) <-> ( X e. B /\ X =/= .0. ) ) )
6 drngring 
 |-  ( R e. DivRing -> R e. Ring )
7 4 3 1 ringinvcl 
 |-  ( ( R e. Ring /\ X e. ( Unit ` R ) ) -> ( I ` X ) e. B )
8 7 ex 
 |-  ( R e. Ring -> ( X e. ( Unit ` R ) -> ( I ` X ) e. B ) )
9 6 8 syl 
 |-  ( R e. DivRing -> ( X e. ( Unit ` R ) -> ( I ` X ) e. B ) )
10 5 9 sylbird 
 |-  ( R e. DivRing -> ( ( X e. B /\ X =/= .0. ) -> ( I ` X ) e. B ) )
11 10 3impib 
 |-  ( ( R e. DivRing /\ X e. B /\ X =/= .0. ) -> ( I ` X ) e. B )