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Theorem slwpgp 17968
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 2621 . . 3 𝐻 = 𝐻
2 slwsubg 17965 . . . 4 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
3 slwpgp.1 . . . . 5 𝑆 = (𝐺s 𝐻)
43slwispgp 17966 . . . 4 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((𝐻𝐻𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻))
52, 4mpdan 701 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) → ((𝐻𝐻𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻))
61, 5mpbiri 248 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝐻𝐻𝑃 pGrp 𝑆))
76simprd 479 1 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3560   class class class wbr 4623  cfv 5857  (class class class)co 6615  s cress 15801  SubGrpcsubg 17528   pGrp cpgp 17886   pSyl cslw 17887
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-subg 17531  df-slw 17891
This theorem is referenced by:  slwhash  17979  sylow2  17981  sylow3lem6  17987
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