Metamath Proof Explorer


Definition df-zrng

Description: Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014)

Ref Expression
Assertion df-zrng ZRing = ( 𝑟 ∈ V ↦ ( 𝑟 IntgRing ran ( ℤRHom ‘ 𝑟 ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 czr ZRing
1 vr 𝑟
2 cvv V
3 1 cv 𝑟
4 citr IntgRing
5 czrh ℤRHom
6 3 5 cfv ( ℤRHom ‘ 𝑟 )
7 6 crn ran ( ℤRHom ‘ 𝑟 )
8 3 7 4 co ( 𝑟 IntgRing ran ( ℤRHom ‘ 𝑟 ) )
9 1 2 8 cmpt ( 𝑟 ∈ V ↦ ( 𝑟 IntgRing ran ( ℤRHom ‘ 𝑟 ) ) )
10 0 9 wceq ZRing = ( 𝑟 ∈ V ↦ ( 𝑟 IntgRing ran ( ℤRHom ‘ 𝑟 ) ) )