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	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
Assertion	rngass	$⊢ (R \in Rng \land (X \in B \land Y \in B \land Z \in B)) \to (X \underset{˙}{\cdot} Y) \underset{˙}{\cdot} Z = X \underset{˙}{\cdot} (Y \underset{˙}{\cdot} Z)$

Step	Hyp	Ref	Expression
1		rngass.b	$⊢ B = {Base}_{R}$
2		rngass.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
3		eqid	$⊢ {mulGrp}_{R} = {mulGrp}_{R}$
4	3	rngmgp	$⊢ R \in Rng \to {mulGrp}_{R} \in Smgrp$
5	3 1	mgpbas	$⊢ B = {Base}_{{mulGrp}_{R}}$
6	3 2	mgpplusg	$⊢ \underset{˙}{\cdot} = +_{{mulGrp}_{R}}$
7	5 6	sgrpass	$⊢ ({mulGrp}_{R} \in Smgrp \land (X \in B \land Y \in B \land Z \in B)) \to (X \underset{˙}{\cdot} Y) \underset{˙}{\cdot} Z = X \underset{˙}{\cdot} (Y \underset{˙}{\cdot} Z)$
8	4 7	sylan	$⊢ (R \in Rng \land (X \in B \land Y \in B \land Z \in B)) \to (X \underset{˙}{\cdot} Y) \underset{˙}{\cdot} Z = X \underset{˙}{\cdot} (Y \underset{˙}{\cdot} Z)$

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