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Definition df-slw 17891
Description: Define the set of Sylow p-subgroups of a group 𝑔. A Sylow p-subgroup is a p-group that is not a subgroup of any other p-groups in 𝑔. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
df-slw pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
Distinct variable group:   𝑔,,𝑘,𝑝

Detailed syntax breakdown of Definition df-slw
StepHypRef Expression
1 cslw 17887 . 2 class pSyl
2 vp . . 3 setvar 𝑝
3 vg . . 3 setvar 𝑔
4 cprime 15328 . . 3 class
5 cgrp 17362 . . 3 class Grp
6 vh . . . . . . . . 9 setvar
76cv 1479 . . . . . . . 8 class
8 vk . . . . . . . . 9 setvar 𝑘
98cv 1479 . . . . . . . 8 class 𝑘
107, 9wss 3560 . . . . . . 7 wff 𝑘
112cv 1479 . . . . . . . 8 class 𝑝
123cv 1479 . . . . . . . . 9 class 𝑔
13 cress 15801 . . . . . . . . 9 class s
1412, 9, 13co 6615 . . . . . . . 8 class (𝑔s 𝑘)
15 cpgp 17886 . . . . . . . 8 class pGrp
1611, 14, 15wbr 4623 . . . . . . 7 wff 𝑝 pGrp (𝑔s 𝑘)
1710, 16wa 384 . . . . . 6 wff (𝑘𝑝 pGrp (𝑔s 𝑘))
186, 8weq 1871 . . . . . 6 wff = 𝑘
1917, 18wb 196 . . . . 5 wff ((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)
20 csubg 17528 . . . . . 6 class SubGrp
2112, 20cfv 5857 . . . . 5 class (SubGrp‘𝑔)
2219, 8, 21wral 2908 . . . 4 wff 𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)
2322, 6, 21crab 2912 . . 3 class { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)}
242, 3, 4, 5, 23cmpt2 6617 . 2 class (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
251, 24wceq 1480 1 wff pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
Colors of variables: wff setvar class
This definition is referenced by:  isslw  17963
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