Theorem ispgp 19522

Description: A group is a 𝑃-group if every element has some power of 𝑃 as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

Ref	Expression
ispgp.1	⊢ 𝑋 = (Base‘𝐺)
ispgp.2	⊢ 𝑂 = (od‘𝐺)

Assertion

Ref	Expression
ispgp	⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))

Distinct variable groups:   𝑥,𝑛,𝐺   𝑃,𝑛,𝑥   𝑥,𝑋

Allowed substitution hints:   𝑂(𝑥,𝑛)   𝑋(𝑛)

Proof of Theorem ispgp

Dummy variables 𝑔 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
2	1	fveq2d 6862	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
3		ispgp.1	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
4	2, 3	eqtr4di 2782	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
5	1	fveq2d 6862	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (od‘𝑔) = (od‘𝐺))
6		ispgp.2	. . . . . . . 8 ⊢ 𝑂 = (od‘𝐺)
7	5, 6	eqtr4di 2782	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (od‘𝑔) = 𝑂)
8	7	fveq1d 6860	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → ((od‘𝑔)‘𝑥) = (𝑂‘𝑥))
9		simpl 482	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑝 = 𝑃)
10	9	oveq1d 7402	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑝↑𝑛) = (𝑃↑𝑛))
11	8, 10	eqeq12d 2745	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (((od‘𝑔)‘𝑥) = (𝑝↑𝑛) ↔ (𝑂‘𝑥) = (𝑃↑𝑛)))
12	11	rexbidv 3157	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
13	4, 12	raleqbidv 3319	. . 3 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛) ↔ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
14		df-pgp 19460	. . 3 ⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
15	13, 14	brab2a 5732	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
16		df-3an 1088	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)) ↔ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
17	15, 16	bitr4i 278	1 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  ℕ₀cn0 12442  ↑cexp 14026  ℙcprime 16641  Basecbs 17179  Grpcgrp 18865  odcod 19454   pGrp cpgp 19456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-iota 6464  df-fv 6519  df-ov 7390  df-pgp 19460

This theorem is referenced by:  pgpprm  19523  pgpgrp  19524  pgpfi1  19525  subgpgp  19527  pgpfi  19535