Theorem pgpprm 18710

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 2798	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2798	. . 3 ⊢ (od‘𝐺) = (od‘𝐺)
3	1, 2	ispgp 18709	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛)))
4	3	simp1bi 1142	1 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  ℕ₀cn0 11885  ↑cexp 13425  ℙcprime 16005  Basecbs 16475  Grpcgrp 18095  odcod 18644   pGrp cpgp 18646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-iota 6283  df-fv 6332  df-ov 7138  df-pgp 18650

This theorem is referenced by:  subgpgp  18714  pgpssslw  18731  sylow2blem3  18739  pgpfac1lem2  19190  pgpfac1lem3a  19191  pgpfac1lem3  19192  pgpfac1lem4  19193  pgpfaclem1  19196