iscrng2

Description: A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015)

	Ref	Expression
Hypotheses	ringcl.b	$⊢ B = {Base}_{R}$
	ringcl.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
Assertion	iscrng2	$⊢ R \in CRing \leftrightarrow (R \in Ring \land \forall x \in B \forall y \in B x \underset{˙}{\cdot} y = y \underset{˙}{\cdot} x)$

Step	Hyp	Ref	Expression
1		ringcl.b	$⊢ B = {Base}_{R}$
2		ringcl.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
3		eqid	$⊢ {mulGrp}_{R} = {mulGrp}_{R}$
4	3	iscrng	$⊢ R \in CRing \leftrightarrow (R \in Ring \land {mulGrp}_{R} \in CMnd)$
5	3	ringmgp	$⊢ R \in Ring \to {mulGrp}_{R} \in Mnd$
6	3 1	mgpbas	$⊢ B = {Base}_{{mulGrp}_{R}}$
7	3 2	mgpplusg	$⊢ \underset{˙}{\cdot} = +_{{mulGrp}_{R}}$
8	6 7	iscmn	$⊢ {mulGrp}_{R} \in CMnd \leftrightarrow ({mulGrp}_{R} \in Mnd \land \forall x \in B \forall y \in B x \underset{˙}{\cdot} y = y \underset{˙}{\cdot} x)$
9	8	baib	$⊢ {mulGrp}_{R} \in Mnd \to ({mulGrp}_{R} \in CMnd \leftrightarrow \forall x \in B \forall y \in B x \underset{˙}{\cdot} y = y \underset{˙}{\cdot} x)$
10	5 9	syl	$⊢ R \in Ring \to ({mulGrp}_{R} \in CMnd \leftrightarrow \forall x \in B \forall y \in B x \underset{˙}{\cdot} y = y \underset{˙}{\cdot} x)$
11	10	pm5.32i	$⊢ (R \in Ring \land {mulGrp}_{R} \in CMnd) \leftrightarrow (R \in Ring \land \forall x \in B \forall y \in B x \underset{˙}{\cdot} y = y \underset{˙}{\cdot} x)$
12	4 11	bitri	$⊢ R \in CRing \leftrightarrow (R \in Ring \land \forall x \in B \forall y \in B x \underset{˙}{\cdot} y = y \underset{˙}{\cdot} x)$

Metamath Proof Explorer