Metamath Proof Explorer


Theorem ringcl

Description: Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

Ref Expression
Hypotheses ringcl.b
|- B = ( Base ` R )
ringcl.t
|- .x. = ( .r ` R )
Assertion ringcl
|- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )

Proof

Step Hyp Ref Expression
1 ringcl.b
 |-  B = ( Base ` R )
2 ringcl.t
 |-  .x. = ( .r ` R )
3 eqid
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
4 3 ringmgp
 |-  ( R e. Ring -> ( mulGrp ` R ) e. Mnd )
5 3 1 mgpbas
 |-  B = ( Base ` ( mulGrp ` R ) )
6 3 2 mgpplusg
 |-  .x. = ( +g ` ( mulGrp ` R ) )
7 5 6 mndcl
 |-  ( ( ( mulGrp ` R ) e. Mnd /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
8 4 7 syl3an1
 |-  ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )