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Mirrors > Home > MPE Home > Th. List > df-subg | Structured version Visualization version GIF version |
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 18234), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 18229), contains the neutral element of the group (see subg0 18225) and contains the inverses for all of its elements (see subginvcl 18228). (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
df-subg | ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csubg 18213 | . 2 class SubGrp | |
2 | vw | . . 3 setvar 𝑤 | |
3 | cgrp 18043 | . . 3 class Grp | |
4 | 2 | cv 1527 | . . . . . 6 class 𝑤 |
5 | vs | . . . . . . 7 setvar 𝑠 | |
6 | 5 | cv 1527 | . . . . . 6 class 𝑠 |
7 | cress 16474 | . . . . . 6 class ↾s | |
8 | 4, 6, 7 | co 7145 | . . . . 5 class (𝑤 ↾s 𝑠) |
9 | 8, 3 | wcel 2105 | . . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp |
10 | cbs 16473 | . . . . . 6 class Base | |
11 | 4, 10 | cfv 6349 | . . . . 5 class (Base‘𝑤) |
12 | 11 | cpw 4537 | . . . 4 class 𝒫 (Base‘𝑤) |
13 | 9, 5, 12 | crab 3142 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp} |
14 | 2, 3, 13 | cmpt 5138 | . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}) |
15 | 1, 14 | wceq 1528 | 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}) |
Colors of variables: wff setvar class |
This definition is referenced by: issubg 18219 |
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