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Theorem slwpgp 19646
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 2735 . . 3 𝐻 = 𝐻
2 slwsubg 19643 . . . 4 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
3 slwpgp.1 . . . . 5 𝑆 = (𝐺s 𝐻)
43slwispgp 19644 . . . 4 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((𝐻𝐻𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻))
52, 4mpdan 687 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) → ((𝐻𝐻𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻))
61, 5mpbiri 258 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝐻𝐻𝑃 pGrp 𝑆))
76simprd 495 1 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wss 3963   class class class wbr 5148  cfv 6563  (class class class)co 7431  s cress 17274  SubGrpcsubg 19151   pGrp cpgp 19559   pSyl cslw 19560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-subg 19154  df-slw 19564
This theorem is referenced by:  slwhash  19657  sylow2  19659  sylow3lem6  19665
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