Theorem zncrng2

Description: Making a commutative ring as a quotient of ZZ and n ZZ . (Contributed by Mario Carneiro, 12-Jun-2015) (Revised by AV, 13-Jun-2019)

	Ref	Expression
Hypotheses	znval.s	⊢ 𝑆 = ( RSpan ‘ ℤring )
	znval.u	⊢ 𝑈 = ( ℤring /s ( ℤring ~QG ( 𝑆 ‘ { 𝑁 } ) ) )
Assertion	zncrng2	⊢ ( 𝑁 ∈ ℤ → 𝑈 ∈ CRing )

Proof

Step	Hyp	Ref	Expression
1		znval.s	⊢ 𝑆 = ( RSpan ‘ ℤring )
2		znval.u	⊢ 𝑈 = ( ℤring /s ( ℤring ~QG ( 𝑆 ‘ { 𝑁 } ) ) )
3		zringcrng	⊢ ℤring ∈ CRing
4	1	znlidl	⊢ ( 𝑁 ∈ ℤ → ( 𝑆 ‘ { 𝑁 } ) ∈ ( LIdeal ‘ ℤring ) )
5		eqid	⊢ ( LIdeal ‘ ℤring ) = ( LIdeal ‘ ℤring )
6	2 5	quscrng	⊢ ( ( ℤring ∈ CRing ∧ ( 𝑆 ‘ { 𝑁 } ) ∈ ( LIdeal ‘ ℤring ) ) → 𝑈 ∈ CRing )
7	3 4 6	sylancr	⊢ ( 𝑁 ∈ ℤ → 𝑈 ∈ CRing )