Theorem pgpprm 19664

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 2763	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2763	. . 3 ⊢ (od‘𝐺) = (od‘𝐺)
3	1, 2	ispgp 19663	. 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)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  ℕ₀cn0 12505  ↑cexp 14099  ℙcprime 16730  Basecbs 17270  Grpcgrp 19001  odcod 19595   pGrp cpgp 19597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-iota 6494  df-fv 6546  df-ov 7415  df-pgp 19601

This theorem is referenced by:  subgpgp  19668  pgpssslw  19685  sylow2blem3  19693  pgpfac1lem2  20148  pgpfac1lem3a  20149  pgpfac1lem3  20150  pgpfac1lem4  20151  pgpfaclem1  20154