Theorem pgpgrp 19627

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 2735	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2735	. . 3 ⊢ (od‘𝐺) = (od‘𝐺)
3	1, 2	ispgp 19625	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛)))
4	3	simp2bi 1145	1 ⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  ℕ₀cn0 12524  ↑cexp 14099  ℙcprime 16705  Basecbs 17245  Grpcgrp 18964  odcod 19557   pGrp cpgp 19559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-iota 6516  df-fv 6571  df-ov 7434  df-pgp 19563

This theorem is referenced by:  pgphash  19640