This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ⁡ ℤ ring
znval.u ⊢ U = ℤ ring / 𝑠 ℤ ring ~ QG S ⁡ N
Assertion zncrng2 ⊢ N ∈ ℤ → U ∈ CRing

Proof

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