Theorem pgpprm 19616

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 2761	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2761	. . 3 ⊢ (od‘𝐺) = (od‘𝐺)
3	1, 2	ispgp 19615	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛)))
4	3	simp1bi 1157	1 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392  ℕ₀cn0 12478  ↑cexp 14071  ℙcprime 16688  Basecbs 17228  Grpcgrp 18958  odcod 19547   pGrp cpgp 19549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-iota 6473  df-fv 6525  df-ov 7395  df-pgp 19553

This theorem is referenced by:  subgpgp  19620  pgpssslw  19637  sylow2blem3  19645  pgpfac1lem2  20100  pgpfac1lem3a  20101  pgpfac1lem3  20102  pgpfac1lem4  20103  pgpfaclem1  20106