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Theorem pgpgrp 19462
Description: Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
Assertion
Ref Expression
pgpgrp (𝑃 pGrp 𝐺𝐺 ∈ Grp)

Proof of Theorem pgpgrp
Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2733 . . 3 (od‘𝐺) = (od‘𝐺)
31, 2ispgp 19460 . 2 (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛)))
43simp2bi 1147 1 (𝑃 pGrp 𝐺𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3062  wrex 3071   class class class wbr 5149  cfv 6544  (class class class)co 7409  0cn0 12472  cexp 14027  cprime 16608  Basecbs 17144  Grpcgrp 18819  odcod 19392   pGrp cpgp 19394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-iota 6496  df-fv 6552  df-ov 7412  df-pgp 19398
This theorem is referenced by:  pgphash  19475
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