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Theorem slwsubg 18006
Description: A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
slwsubg (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))

Proof of Theorem slwsubg
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 isslw 18004 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
21simp2bi 1075 1 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  wss 3567   class class class wbr 4644  cfv 5876  (class class class)co 6635  cprime 15366  s cress 15839  SubGrpcsubg 17569   pGrp cpgp 17927   pSyl cslw 17928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-subg 17572  df-slw 17932
This theorem is referenced by:  slwpgp  18009  subgslw  18012  slwhash  18020  fislw  18021  sylow2  18022  sylow3lem1  18023  sylow3lem2  18024  sylow3lem3  18025  sylow3lem4  18026  sylow3lem5  18027  sylow3lem6  18028
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