Theorem pgpprm 19566

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 2740	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2740	. . 3 ⊢ (od‘𝐺) = (od‘𝐺)
3	1, 2	ispgp 19565	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛)))
4	3	simp1bi 1151	1 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  ℕ₀cn0 12435  ↑cexp 14021  ℙcprime 16638  Basecbs 17177  Grpcgrp 18907  odcod 19497   pGrp cpgp 19499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-iota 6448  df-fv 6500  df-ov 7366  df-pgp 19503

This theorem is referenced by:  subgpgp  19570  pgpssslw  19587  sylow2blem3  19595  pgpfac1lem2  20050  pgpfac1lem3a  20051  pgpfac1lem3  20052  pgpfac1lem4  20053  pgpfaclem1  20056