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Theorem isslw 18223
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 18151 . . 3 pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
21elmpt2cl 7041 . 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 17800 . . . 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 6356 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (SubGrp‘𝑔) = (SubGrp‘𝐺))
9 simpl 474 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑝 = 𝑃)
107oveq1d 6828 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑔s 𝑘) = (𝐺s 𝑘))
119, 10breq12d 4817 . . . . . . . . . . 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 3291 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)))
158, 14rabeqbidv 3335 . . . . . . 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 6362 . . . . . . . 8 (SubGrp‘𝐺) ∈ V
1716rabex 4964 . . . . . . 7 { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)} ∈ V
1815, 1, 17ovmpt2a 6956 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝑃 pSyl 𝐺) = { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)})
1918eleq2d 2825 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ 𝐻 ∈ { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)}))
20 sseq1 3767 . . . . . . . . 9 ( = 𝐻 → (𝑘𝐻𝑘))
2120anbi1d 743 . . . . . . . 8 ( = 𝐻 → ((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
22 eqeq1 2764 . . . . . . . 8 ( = 𝐻 → ( = 𝑘𝐻 = 𝑘))
2321, 22bibi12d 334 . . . . . . 7 ( = 𝐻 → (((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2423ralbidv 3124 . . . . . 6 ( = 𝐻 → (∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2524elrab 3504 . . . . 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 2139  wral 3050  {crab 3054  wss 3715   class class class wbr 4804  cfv 6049  (class class class)co 6813  cprime 15587  s cress 16060  Grpcgrp 17623  SubGrpcsubg 17789   pGrp cpgp 18146   pSyl cslw 18147
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 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
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 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-subg 17792  df-slw 18151
This theorem is referenced by:  slwprm  18224  slwsubg  18225  slwispgp  18226  pgpssslw  18229  subgslw  18231  fislw  18240
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