This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem rngass

Description: Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011) (Revised by AV, 13-Feb-2025)

Ref Expression
Hypotheses rngass.b 
|- B = ( Base ` R )
rngass.t 
|- .x. = ( .r ` R )
Assertion rngass 
|- ( ( R e. Rng /\ ( 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 rngass.b 
 |-  B = ( Base ` R )
2 rngass.t 
 |-  .x. = ( .r ` R )
3 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
4 3 rngmgp 
 |-  ( R e. Rng -> ( mulGrp ` R ) e. Smgrp )
5 3 1 mgpbas 
 |-  B = ( Base ` ( mulGrp ` R ) )
6 3 2 mgpplusg 
 |-  .x. = ( +g ` ( mulGrp ` R ) )
7 5 6 sgrpass 
 |-  ( ( ( mulGrp ` R ) e. Smgrp /\ ( 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. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )