Theorem pgpprm 19812

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 2760	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2760	. . 3 ⊢ (od‘𝐺) = (od‘𝐺)
3	1, 2	ispgp 19811	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  ℕ₀cn0 12609  ↑cexp 14207  ℙcprime 16851  Basecbs 17392  Grpcgrp 19149  odcod 19743   pGrp cpgp 19745

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-iota 6494  df-fv 6546  df-ov 7423  df-pgp 19749

This theorem is used by:  subgpgp  19816  pgpssslw  19833  sylow2blem3  19841  pgpfac1lem2  20296  pgpfac1lem3a  20297  pgpfac1lem3  20298  pgpfac1lem4  20299  pgpfaclem1  20302