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Theorem isslw 18244
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 18172 . . 3 pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
21elmpt2cl 7043 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
3 simp1 1131 . . 3 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝑃 ∈ ℙ)
4 subgrcl 17821 . . . 4 (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
543ad2ant2 1129 . . 3 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝐺 ∈ Grp)
63, 5jca 555 . 2 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
7 simpr 479 . . . . . . . . 9 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑔 = 𝐺)
87fveq2d 6358 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (SubGrp‘𝑔) = (SubGrp‘𝐺))
9 simpl 474 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑝 = 𝑃)
107oveq1d 6830 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑔s 𝑘) = (𝐺s 𝑘))
119, 10breq12d 4818 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑝 pGrp (𝑔s 𝑘) ↔ 𝑃 pGrp (𝐺s 𝑘)))
1211anbi2d 742 . . . . . . . . . 10 ((𝑝 = 𝑃𝑔 = 𝐺) → ((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ (𝑘𝑃 pGrp (𝐺s 𝑘))))
1312bibi1d 332 . . . . . . . . 9 ((𝑝 = 𝑃𝑔 = 𝐺) → (((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘) ↔ ((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)))
148, 13raleqbidv 3292 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)))
158, 14rabeqbidv 3336 . . . . . . 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 6364 . . . . . . . 8 (SubGrp‘𝐺) ∈ V
1716rabex 4965 . . . . . . 7 { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)} ∈ V
1815, 1, 17ovmpt2a 6958 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝑃 pSyl 𝐺) = { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)})
1918eleq2d 2826 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ 𝐻 ∈ { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)}))
20 sseq1 3768 . . . . . . . . 9 ( = 𝐻 → (𝑘𝐻𝑘))
2120anbi1d 743 . . . . . . . 8 ( = 𝐻 → ((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
22 eqeq1 2765 . . . . . . . 8 ( = 𝐻 → ( = 𝑘𝐻 = 𝑘))
2321, 22bibi12d 334 . . . . . . 7 ( = 𝐻 → (((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2423ralbidv 3125 . . . . . 6 ( = 𝐻 → (∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2524elrab 3505 . . . . 5 (𝐻 ∈ { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)} ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2619, 25syl6bb 276 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))))
27 simpl 474 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → 𝑃 ∈ ℙ)
2827biantrurd 530 . . . 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)))))
2926, 28bitrd 268 . . 3 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))))
30 3anass 1081 . . 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))))
3129, 30syl6bbr 278 . 2 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))))
322, 6, 31pm5.21nii 367 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2140  wral 3051  {crab 3055  wss 3716   class class class wbr 4805  cfv 6050  (class class class)co 6815  cprime 15608  s cress 16081  Grpcgrp 17644  SubGrpcsubg 17810   pGrp cpgp 18167   pSyl cslw 18168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-subg 17813  df-slw 18172
This theorem is referenced by:  slwprm  18245  slwsubg  18246  slwispgp  18247  pgpssslw  18250  subgslw  18252  fislw  18261
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