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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  ℕ0cn0 11169  ↑cexp 12722  ℙcprime 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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