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Theorem isslw 17944
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 17872 . . 3 pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
21elmpt2cl 6829 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
3 simp1 1059 . . 3 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝑃 ∈ ℙ)
4 subgrcl 17520 . . . 4 (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
543ad2ant2 1081 . . 3 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝐺 ∈ Grp)
63, 5jca 554 . 2 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
7 simpr 477 . . . . . . . . 9 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑔 = 𝐺)
87fveq2d 6152 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (SubGrp‘𝑔) = (SubGrp‘𝐺))
9 simpl 473 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑝 = 𝑃)
107oveq1d 6619 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑔s 𝑘) = (𝐺s 𝑘))
119, 10breq12d 4626 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑝 pGrp (𝑔s 𝑘) ↔ 𝑃 pGrp (𝐺s 𝑘)))
1211anbi2d 739 . . . . . . . . . 10 ((𝑝 = 𝑃𝑔 = 𝐺) → ((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ (𝑘𝑃 pGrp (𝐺s 𝑘))))
1312bibi1d 333 . . . . . . . . 9 ((𝑝 = 𝑃𝑔 = 𝐺) → (((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘) ↔ ((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)))
148, 13raleqbidv 3141 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)))
158, 14rabeqbidv 3181 . . . . . . 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 6158 . . . . . . . 8 (SubGrp‘𝐺) ∈ V
1716rabex 4773 . . . . . . 7 { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)} ∈ V
1815, 1, 17ovmpt2a 6744 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝑃 pSyl 𝐺) = { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)})
1918eleq2d 2684 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ 𝐻 ∈ { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)}))
20 sseq1 3605 . . . . . . . . 9 ( = 𝐻 → (𝑘𝐻𝑘))
2120anbi1d 740 . . . . . . . 8 ( = 𝐻 → ((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
22 eqeq1 2625 . . . . . . . 8 ( = 𝐻 → ( = 𝑘𝐻 = 𝑘))
2321, 22bibi12d 335 . . . . . . 7 ( = 𝐻 → (((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2423ralbidv 2980 . . . . . 6 ( = 𝐻 → (∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2524elrab 3346 . . . . 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 473 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → 𝑃 ∈ ℙ)
2827biantrurd 529 . . . 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 1040 . . 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 368 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  wss 3555   class class class wbr 4613  cfv 5847  (class class class)co 6604  cprime 15309  s cress 15782  Grpcgrp 17343  SubGrpcsubg 17509   pGrp cpgp 17867   pSyl cslw 17868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-subg 17512  df-slw 17872
This theorem is referenced by:  slwprm  17945  slwsubg  17946  slwispgp  17947  pgpssslw  17950  subgslw  17952  fislw  17961
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