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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-zrng | Structured version Visualization version GIF version |
Description: Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
df-zrng | ⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | czr 34632 | . 2 class ZRing | |
2 | vr | . . 3 setvar 𝑟 | |
3 | cvv 3475 | . . 3 class V | |
4 | 2 | cv 1541 | . . . 4 class 𝑟 |
5 | czrh 21049 | . . . . . 6 class ℤRHom | |
6 | 4, 5 | cfv 6544 | . . . . 5 class (ℤRHom‘𝑟) |
7 | 6 | crn 5678 | . . . 4 class ran (ℤRHom‘𝑟) |
8 | cirng 32747 | . . . 4 class IntgRing | |
9 | 4, 7, 8 | co 7409 | . . 3 class (𝑟 IntgRing ran (ℤRHom‘𝑟)) |
10 | 2, 3, 9 | cmpt 5232 | . 2 class (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟))) |
11 | 1, 10 | wceq 1542 | 1 wff ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟))) |
Colors of variables: wff setvar class |
This definition is referenced by: (None) |
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