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Metamath Proof Explorer

Theorem rngcl

Description: Closure of the multiplication operation of a non-unital ring. (Contributed by AV, 17-Apr-2020)

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

Proof

Step Hyp Ref Expression
1 rngcl.b 
 |-  B = ( Base ` R )
2 rngcl.t 
 |-  .x. = ( .r ` R )
3 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
4 3 rngmgp 
 |-  ( R e. Rng -> ( mulGrp ` R ) e. Smgrp )
5 sgrpmgm 
 |-  ( ( mulGrp ` R ) e. Smgrp -> ( mulGrp ` R ) e. Mgm )
6 4 5 syl 
 |-  ( R e. Rng -> ( mulGrp ` R ) e. Mgm )
7 3 1 mgpbas 
 |-  B = ( Base ` ( mulGrp ` R ) )
8 3 2 mgpplusg 
 |-  .x. = ( +g ` ( mulGrp ` R ) )
9 7 8 mgmcl 
 |-  ( ( ( mulGrp ` R ) e. Mgm /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
10 6 9 syl3an1 
 |-  ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )