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Theorem isslw 18854
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 18780 . . 3 pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
21elmpocl 7406 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
3 simp1 1137 . . 3 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝑃 ∈ ℙ)
4 subgrcl 18405 . . . 4 (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
543ad2ant2 1135 . . 3 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝐺 ∈ Grp)
63, 5jca 515 . 2 ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
7 simpr 488 . . . . . . . . 9 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑔 = 𝐺)
87fveq2d 6681 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (SubGrp‘𝑔) = (SubGrp‘𝐺))
9 simpl 486 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑝 = 𝑃)
107oveq1d 7188 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑔s 𝑘) = (𝐺s 𝑘))
119, 10breq12d 5044 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑝 pGrp (𝑔s 𝑘) ↔ 𝑃 pGrp (𝐺s 𝑘)))
1211anbi2d 632 . . . . . . . . . 10 ((𝑝 = 𝑃𝑔 = 𝐺) → ((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ (𝑘𝑃 pGrp (𝐺s 𝑘))))
1312bibi1d 347 . . . . . . . . 9 ((𝑝 = 𝑃𝑔 = 𝐺) → (((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘) ↔ ((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)))
148, 13raleqbidv 3305 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)))
158, 14rabeqbidv 3388 . . . . . . 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 6690 . . . . . . . 8 (SubGrp‘𝐺) ∈ V
1716rabex 5201 . . . . . . 7 { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)} ∈ V
1815, 1, 17ovmpoa 7323 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝑃 pSyl 𝐺) = { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)})
1918eleq2d 2819 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ 𝐻 ∈ { ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘)}))
20 cleq1lem 14434 . . . . . . . 8 ( = 𝐻 → ((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
21 eqeq1 2743 . . . . . . . 8 ( = 𝐻 → ( = 𝑘𝐻 = 𝑘))
2220, 21bibi12d 349 . . . . . . 7 ( = 𝐻 → (((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2322ralbidv 3110 . . . . . 6 ( = 𝐻 → (∀𝑘 ∈ (SubGrp‘𝐺)((𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
2423elrab 3589 . . . . 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 290 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))))
26 simpl 486 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → 𝑃 ∈ ℙ)
2726biantrurd 536 . . . 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 282 . . 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 1096 . . 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 292 . 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 383 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  {crab 3058  wss 3844   class class class wbr 5031  cfv 6340  (class class class)co 7173  cprime 16115  s cress 16590  Grpcgrp 18222  SubGrpcsubg 18394   pGrp cpgp 18775   pSyl cslw 18776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-subg 18397  df-slw 18780
This theorem is referenced by:  slwprm  18855  slwsubg  18856  slwispgp  18857  pgpssslw  18860  subgslw  18862  fislw  18871
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