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Theorem slwpgp 17851
Description: A Sylow 𝑃-subgroup is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
slwpgp.1 𝑆 = (𝐺s 𝐻)
Assertion
Ref Expression
slwpgp (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆)

Proof of Theorem slwpgp
StepHypRef Expression
1 eqid 2610 . . 3 𝐻 = 𝐻
2 slwsubg 17848 . . . 4 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
3 slwpgp.1 . . . . 5 𝑆 = (𝐺s 𝐻)
43slwispgp 17849 . . . 4 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((𝐻𝐻𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻))
52, 4mpdan 699 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) → ((𝐻𝐻𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻))
61, 5mpbiri 247 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝐻𝐻𝑃 pGrp 𝑆))
76simprd 478 1 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wss 3540   class class class wbr 4583  cfv 5804  (class class class)co 6549  s cress 15696  SubGrpcsubg 17411   pGrp cpgp 17769   pSyl cslw 17770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-subg 17414  df-slw 17774
This theorem is referenced by:  slwhash  17862  sylow2  17864  sylow3lem6  17870
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