![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 19089), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19084), contains the neutral element of the group (see subg0 19080) and contains the inverses for all of its elements (see subginvcl 19083). (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 19068 | . 2 class SubGrp | |
2 | vw | . . 3 setvar 𝑤 | |
3 | cgrp 18883 | . . 3 class Grp | |
4 | 2 | cv 1533 | . . . . . 6 class 𝑤 |
5 | vs | . . . . . . 7 setvar 𝑠 | |
6 | 5 | cv 1533 | . . . . . 6 class 𝑠 |
7 | cress 17202 | . . . . . 6 class ↾s | |
8 | 4, 6, 7 | co 7414 | . . . . 5 class (𝑤 ↾s 𝑠) |
9 | 8, 3 | wcel 2099 | . . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp |
10 | cbs 17173 | . . . . . 6 class Base | |
11 | 4, 10 | cfv 6542 | . . . . 5 class (Base‘𝑤) |
12 | 11 | cpw 4598 | . . . 4 class 𝒫 (Base‘𝑤) |
13 | 9, 5, 12 | crab 3428 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp} |
14 | 2, 3, 13 | cmpt 5225 | . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}) |
15 | 1, 14 | wceq 1534 | 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 19074 |
Copyright terms: Public domain | W3C validator |