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Theorem pgpgrp 19503
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 2730 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2730 . . 3 (od‘𝐺) = (od‘𝐺)
31, 2ispgp 19501 . 2 (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛)))
43simp2bi 1144 1 (𝑃 pGrp 𝐺𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2104  wral 3059  wrex 3068   class class class wbr 5147  cfv 6542  (class class class)co 7411  0cn0 12476  cexp 14031  cprime 16612  Basecbs 17148  Grpcgrp 18855  odcod 19433   pGrp cpgp 19435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-iota 6494  df-fv 6550  df-ov 7414  df-pgp 19439
This theorem is referenced by:  pgphash  19516
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