Theorem pgpprm 19694

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 2766	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2766	. . 3 ⊢ (od‘𝐺) = (od‘𝐺)
3	1, 2	ispgp 19693	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛)))
4	3	simp1bi 1163	1 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  ℕ₀cn0 12522  ↑cexp 14117  ℙcprime 16754  Basecbs 17294  Grpcgrp 19031  odcod 19625   pGrp cpgp 19627

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-iota 6499  df-fv 6551  df-ov 7426  df-pgp 19631

This theorem is used by:  subgpgp  19698  pgpssslw  19715  sylow2blem3  19723  pgpfac1lem2  20178  pgpfac1lem3a  20179  pgpfac1lem3  20180  pgpfac1lem4  20181  pgpfaclem1  20184