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Metamath Proof Explorer

Theorem ringass

Description: Associative law for multiplication in a ring. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

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

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 mndass 
 |-  ( ( ( mulGrp ` R ) e. Mnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )
8 4 7 sylan 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )