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Theorem pgpgrp 18209
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 2760 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2760 . . 3 (od‘𝐺) = (od‘𝐺)
31, 2ispgp 18207 . 2 (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛)))
43simp2bi 1141 1 (𝑃 pGrp 𝐺𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wral 3050  wrex 3051   class class class wbr 4804  cfv 6049  (class class class)co 6813  0cn0 11484  cexp 13054  cprime 15587  Basecbs 16059  Grpcgrp 17623  odcod 18144   pGrp cpgp 18146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-iota 6012  df-fv 6057  df-ov 6816  df-pgp 18150
This theorem is referenced by:  pgphash  18222
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