Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-eqg Structured version   Visualization version   GIF version

Definition df-eqg 17416
 Description: Define the equivalence relation in a quotient ring or quotient group (where 𝑖 is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
Assertion
Ref Expression
df-eqg ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)})
Distinct variable group:   𝑖,𝑟,𝑥,𝑦

Detailed syntax breakdown of Definition df-eqg
StepHypRef Expression
1 cqg 17413 . 2 class ~QG
2 vr . . 3 setvar 𝑟
3 vi . . 3 setvar 𝑖
4 cvv 3173 . . 3 class V
5 vx . . . . . . . 8 setvar 𝑥
65cv 1474 . . . . . . 7 class 𝑥
7 vy . . . . . . . 8 setvar 𝑦
87cv 1474 . . . . . . 7 class 𝑦
96, 8cpr 4127 . . . . . 6 class {𝑥, 𝑦}
102cv 1474 . . . . . . 7 class 𝑟
11 cbs 15695 . . . . . . 7 class Base
1210, 11cfv 5804 . . . . . 6 class (Base‘𝑟)
139, 12wss 3540 . . . . 5 wff {𝑥, 𝑦} ⊆ (Base‘𝑟)
14 cminusg 17246 . . . . . . . . 9 class invg
1510, 14cfv 5804 . . . . . . . 8 class (invg𝑟)
166, 15cfv 5804 . . . . . . 7 class ((invg𝑟)‘𝑥)
17 cplusg 15768 . . . . . . . 8 class +g
1810, 17cfv 5804 . . . . . . 7 class (+g𝑟)
1916, 8, 18co 6549 . . . . . 6 class (((invg𝑟)‘𝑥)(+g𝑟)𝑦)
203cv 1474 . . . . . 6 class 𝑖
2119, 20wcel 1977 . . . . 5 wff (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖
2213, 21wa 383 . . . 4 wff ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)
2322, 5, 7copab 4642 . . 3 class {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)}
242, 3, 4, 4, 23cmpt2 6551 . 2 class (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)})
251, 24wceq 1475 1 wff ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)})
 Colors of variables: wff setvar class This definition is referenced by:  releqg  17464  eqgfval  17465
 Copyright terms: Public domain W3C validator