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Metamath Proof Explorer

Theorem cringm4

Description: Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024)

Ref Expression
Hypotheses cringm4.1 
|- B = ( Base ` R )
cringm4.2 
|- .x. = ( .r ` R )
Assertion cringm4 
|- ( ( R e. CRing /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .x. Y ) .x. ( Z .x. W ) ) = ( ( X .x. Z ) .x. ( Y .x. W ) ) )

Proof

Step Hyp Ref Expression
1 cringm4.1 
 |-  B = ( Base ` R )
2 cringm4.2 
 |-  .x. = ( .r ` R )
3 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
4 3 crngmgp 
 |-  ( R e. CRing -> ( mulGrp ` R ) e. CMnd )
5 3 1 mgpbas 
 |-  B = ( Base ` ( mulGrp ` R ) )
6 3 2 mgpplusg 
 |-  .x. = ( +g ` ( mulGrp ` R ) )
7 5 6 cmn4 
 |-  ( ( ( mulGrp ` R ) e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .x. Y ) .x. ( Z .x. W ) ) = ( ( X .x. Z ) .x. ( Y .x. W ) ) )
8 4 7 syl3an1 
 |-  ( ( R e. CRing /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .x. Y ) .x. ( Z .x. W ) ) = ( ( X .x. Z ) .x. ( Y .x. W ) ) )