Theorem pgpprm 19574

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

Assertion

Ref	Expression
pgpprm	⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)

Proof of Theorem pgpprm

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 19573	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛)))
4	3	simp1bi 1145	1 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  ℕ₀cn0 12501  ↑cexp 14079  ℙcprime 16690  Basecbs 17228  Grpcgrp 18916  odcod 19505   pGrp cpgp 19507

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-iota 6484  df-fv 6539  df-ov 7408  df-pgp 19511

This theorem is referenced by:  subgpgp  19578  pgpssslw  19595  sylow2blem3  19603  pgpfac1lem2  20058  pgpfac1lem3a  20059  pgpfac1lem3  20060  pgpfac1lem4  20061  pgpfaclem1  20064