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Theorem isslw 19213
Description: The property of being a Sylow subgroup. A Sylow 𝑃-subgroup is a 𝑃-group which has no proper supersets that are also 𝑃-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
isslw (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
Distinct variable groups:   𝑘,𝐺   𝑘,𝐻   𝑃,𝑘

Proof of Theorem isslw
Dummy variables 𝑔 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-slw 19139 . . 3 pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
21elmpocl 7511 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
3 simp1 1135 . . 3 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝑃 ∈ ℙ)
4 subgrcl 18760 . . . 4 (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
543ad2ant2 1133 . . 3 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝐺 ∈ Grp)
63, 5jca 512 . 2 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
7 simpr 485 . . . . . . . . 9 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑔 = 𝐺)
87fveq2d 6778 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (SubGrp‘𝑔) = (SubGrp‘𝐺))
9 simpl 483 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑝 = 𝑃)
107oveq1d 7290 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑔s 𝑘) = (𝐺s 𝑘))
119, 10breq12d 5087 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑝 pGrp (𝑔s 𝑘) ↔ 𝑃 pGrp (𝐺s 𝑘)))
1211anbi2d 629 . . . . . . . . . 10 ((𝑝 = 𝑃𝑔 = 𝐺) → ((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ (𝑘𝑃 pGrp (𝐺s 𝑘))))
1312bibi1d 344 . . . . . . . . 9 ((𝑝 = 𝑃𝑔 = 𝐺) → (((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘) ↔ ((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)))
148, 13raleqbidv 3336 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)))
158, 14rabeqbidv 3420 . . . . . . 7 ((𝑝 = 𝑃𝑔 = 𝐺) → { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)} = { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)})
16 fvex 6787 . . . . . . . 8 (SubGrp‘𝐺) ∈ V
1716rabex 5256 . . . . . . 7 { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)} ∈ V
1815, 1, 17ovmpoa 7428 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝑃 pSyl 𝐺) = { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)})
1918eleq2d 2824 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ 𝐻 ∈ { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)}))
20 cleq1lem 14693 . . . . . . . 8 ( = 𝐻 → ((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
21 eqeq1 2742 . . . . . . . 8 ( = 𝐻 → ( = 𝑘𝐻 = 𝑘))
2220, 21bibi12d 346 . . . . . . 7 ( = 𝐻 → (((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2322ralbidv 3112 . . . . . 6 ( = 𝐻 → (∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2423elrab 3624 . . . . 5 (𝐻 ∈ { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)} ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2519, 24bitrdi 287 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))))
26 simpl 483 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → 𝑃 ∈ ℙ)
2726biantrurd 533 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → ((𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) ↔ (𝑃 ∈ ℙ ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))))
2825, 27bitrd 278 . . 3 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))))
29 3anass 1094 . . 3 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) ↔ (𝑃 ∈ ℙ ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))))
3028, 29bitr4di 289 . 2 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))))
312, 6, 30pm5.21nii 380 1 (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  {crab 3068  wss 3887   class class class wbr 5074  cfv 6433  (class class class)co 7275  cprime 16376  s cress 16941  Grpcgrp 18577  SubGrpcsubg 18749   pGrp cpgp 19134   pSyl cslw 19135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-subg 18752  df-slw 19139
This theorem is referenced by:  slwprm  19214  slwsubg  19215  slwispgp  19216  pgpssslw  19219  subgslw  19221  fislw  19230
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