Theorem ispgp 19558

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 6838	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
3		ispgp.1	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
4	2, 3	eqtr4di 2790	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
5	1	fveq2d 6838	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (od‘𝑔) = (od‘𝐺))
6		ispgp.2	. . . . . . . 8 ⊢ 𝑂 = (od‘𝐺)
7	5, 6	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (od‘𝑔) = 𝑂)
8	7	fveq1d 6836	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → ((od‘𝑔)‘𝑥) = (𝑂‘𝑥))
9		simpl 482	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑝 = 𝑃)
10	9	oveq1d 7375	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑝↑𝑛) = (𝑃↑𝑛))
11	8, 10	eqeq12d 2753	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (((od‘𝑔)‘𝑥) = (𝑝↑𝑛) ↔ (𝑂‘𝑥) = (𝑃↑𝑛)))
12	11	rexbidv 3162	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
13	4, 12	raleqbidv 3312	. . 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 19496	. . 3 ⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
15	13, 14	brab2a 5717	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
16		df-3an 1089	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  ℕ₀cn0 12428  ↑cexp 14014  ℙcprime 16631  Basecbs 17170  Grpcgrp 18900  odcod 19490   pGrp cpgp 19492

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-iota 6448  df-fv 6500  df-ov 7363  df-pgp 19496

This theorem is referenced by:  pgpprm  19559  pgpgrp  19560  pgpfi1  19561  subgpgp  19563  pgpfi  19571