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Definition df-irng 33113
 Description: Define the subring of elements of 𝑟 integral over 𝑠 in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
df-irng IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ 𝑓 ∈ (Monic1p‘(𝑟s 𝑠))(𝑓 “ {(0g𝑟)}))
Distinct variable group:   𝑓,𝑟,𝑠

Detailed syntax breakdown of Definition df-irng
StepHypRef Expression
1 citr 33105 . 2 class IntgRing
2 vr . . 3 setvar 𝑟
3 vs . . 3 setvar 𝑠
4 cvv 3409 . . 3 class V
5 vf . . . 4 setvar 𝑓
62cv 1537 . . . . . 6 class 𝑟
73cv 1537 . . . . . 6 class 𝑠
8 cress 16542 . . . . . 6 class s
96, 7, 8co 7150 . . . . 5 class (𝑟s 𝑠)
10 cmn1 24825 . . . . 5 class Monic1p
119, 10cfv 6335 . . . 4 class (Monic1p‘(𝑟s 𝑠))
125cv 1537 . . . . . 6 class 𝑓
1312ccnv 5523 . . . . 5 class 𝑓
14 c0g 16771 . . . . . . 7 class 0g
156, 14cfv 6335 . . . . . 6 class (0g𝑟)
1615csn 4522 . . . . 5 class {(0g𝑟)}
1713, 16cima 5527 . . . 4 class (𝑓 “ {(0g𝑟)})
185, 11, 17ciun 4883 . . 3 class 𝑓 ∈ (Monic1p‘(𝑟s 𝑠))(𝑓 “ {(0g𝑟)})
192, 3, 4, 4, 18cmpo 7152 . 2 class (𝑟 ∈ V, 𝑠 ∈ V ↦ 𝑓 ∈ (Monic1p‘(𝑟s 𝑠))(𝑓 “ {(0g𝑟)}))
201, 19wceq 1538 1 wff IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ 𝑓 ∈ (Monic1p‘(𝑟s 𝑠))(𝑓 “ {(0g𝑟)}))
 Colors of variables: wff setvar class This definition is referenced by: (None)
 Copyright terms: Public domain W3C validator