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	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
Assertion	cringm4	$⊢ (R \in CRing \land (X \in B \land Y \in B) \land (Z \in B \land W \in B)) \to (X \underset{˙}{\cdot} Y) \underset{˙}{\cdot} (Z \underset{˙}{\cdot} W) = (X \underset{˙}{\cdot} Z) \underset{˙}{\cdot} (Y \underset{˙}{\cdot} W)$

Step	Hyp	Ref	Expression
1		cringm4.1	$⊢ B = {Base}_{R}$
2		cringm4.2	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
3		eqid	$⊢ {mulGrp}_{R} = {mulGrp}_{R}$
4	3	crngmgp	$⊢ R \in CRing \to {mulGrp}_{R} \in CMnd$
5	3 1	mgpbas	$⊢ B = {Base}_{{mulGrp}_{R}}$
6	3 2	mgpplusg	$⊢ \underset{˙}{\cdot} = +_{{mulGrp}_{R}}$
7	5 6	cmn4	$⊢ ({mulGrp}_{R} \in CMnd \land (X \in B \land Y \in B) \land (Z \in B \land W \in B)) \to (X \underset{˙}{\cdot} Y) \underset{˙}{\cdot} (Z \underset{˙}{\cdot} W) = (X \underset{˙}{\cdot} Z) \underset{˙}{\cdot} (Y \underset{˙}{\cdot} W)$
8	4 7	syl3an1	$⊢ (R \in CRing \land (X \in B \land Y \in B) \land (Z \in B \land W \in B)) \to (X \underset{˙}{\cdot} Y) \underset{˙}{\cdot} (Z \underset{˙}{\cdot} W) = (X \underset{˙}{\cdot} Z) \underset{˙}{\cdot} (Y \underset{˙}{\cdot} W)$

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