Theorem crngbascntr

Description: The base set of a commutative ring is its center. (Contributed by SN, 21-Mar-2025)

	Ref	Expression
Hypotheses	crngbascntr.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	crngbascntr.z	⊢ 𝑍 = ( Cntr ‘ ( mulGrp ‘ 𝐺 ) )
Assertion	crngbascntr	⊢ ( 𝐺 ∈ CRing → 𝐵 = 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		crngbascntr.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		crngbascntr.z	⊢ 𝑍 = ( Cntr ‘ ( mulGrp ‘ 𝐺 ) )
3		eqid	⊢ ( mulGrp ‘ 𝐺 ) = ( mulGrp ‘ 𝐺 )
4	3	crngmgp	⊢ ( 𝐺 ∈ CRing → ( mulGrp ‘ 𝐺 ) ∈ CMnd )
5	3 1	mgpbas	⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝐺 ) )
6	5 2	cmnbascntr	⊢ ( ( mulGrp ‘ 𝐺 ) ∈ CMnd → 𝐵 = 𝑍 )
7	4 6	syl	⊢ ( 𝐺 ∈ CRing → 𝐵 = 𝑍 )