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Theorem pgpprm 18709
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 𝑥 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2822 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2822 . . 3 (od‘𝐺) = (od‘𝐺)
31, 2ispgp 18708 . 2 (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛)))
43simp1bi 1142 1 (𝑃 pGrp 𝐺𝑃 ∈ ℙ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  wral 3130  wrex 3131   class class class wbr 5042  cfv 6334  (class class class)co 7140  0cn0 11885  cexp 13425  cprime 16004  Basecbs 16474  Grpcgrp 18094  odcod 18643   pGrp cpgp 18645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-iota 6293  df-fv 6342  df-ov 7143  df-pgp 18649
This theorem is referenced by:  subgpgp  18713  pgpssslw  18730  sylow2blem3  18738  pgpfac1lem2  19188  pgpfac1lem3a  19189  pgpfac1lem3  19190  pgpfac1lem4  19191  pgpfaclem1  19194
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