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Theorem pgpprm 17831
Description: Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
Assertion
Ref Expression
pgpprm (𝑃 pGrp 𝐺𝑃 ∈ ℙ)

Proof of Theorem pgpprm
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2610 . . 3 (od‘𝐺) = (od‘𝐺)
31, 2ispgp 17830 . 2 (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛)))
43simp1bi 1069 1 (𝑃 pGrp 𝐺𝑃 ∈ ℙ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wral 2896  wrex 2897   class class class wbr 4583  cfv 5804  (class class class)co 6549  0cn0 11169  cexp 12722  cprime 15223  Basecbs 15695  Grpcgrp 17245  odcod 17767   pGrp cpgp 17769
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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-iota 5768  df-fv 5812  df-ov 6552  df-pgp 17773
This theorem is referenced by:  subgpgp  17835  pgpssslw  17852  sylow2blem3  17860  pgpfac1lem2  18297  pgpfac1lem3a  18298  pgpfac1lem3  18299  pgpfac1lem4  18300  pgpfaclem1  18303
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