Theorem pgpgrp 19460

Description: Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)

Assertion

Ref	Expression
pgpgrp	⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp)

Proof of Theorem pgpgrp

Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2729	. . 3 ⊢ (od‘𝐺) = (od‘𝐺)
3	1, 2	ispgp 19458	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛)))
4	3	simp2bi 1146	1 ⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   class class class wbr 5088  ‘cfv 6476  (class class class)co 7340  ℕ₀cn0 12372  ↑cexp 13956  ℙcprime 16569  Basecbs 17107  Grpcgrp 18799  odcod 19390   pGrp cpgp 19392

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pr 5367

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5089  df-opab 5151  df-xp 5619  df-iota 6432  df-fv 6484  df-ov 7343  df-pgp 19396

This theorem is referenced by:  pgphash  19473