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 ) ) ) |
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 ) ) ) |