Theorem ispgp 19625

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 6911	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
3		ispgp.1	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
4	2, 3	eqtr4di 2793	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
5	1	fveq2d 6911	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (od‘𝑔) = (od‘𝐺))
6		ispgp.2	. . . . . . . 8 ⊢ 𝑂 = (od‘𝐺)
7	5, 6	eqtr4di 2793	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (od‘𝑔) = 𝑂)
8	7	fveq1d 6909	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → ((od‘𝑔)‘𝑥) = (𝑂‘𝑥))
9		simpl 482	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑝 = 𝑃)
10	9	oveq1d 7446	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑝↑𝑛) = (𝑃↑𝑛))
11	8, 10	eqeq12d 2751	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (((od‘𝑔)‘𝑥) = (𝑝↑𝑛) ↔ (𝑂‘𝑥) = (𝑃↑𝑛)))
12	11	rexbidv 3177	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
13	4, 12	raleqbidv 3344	. . 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 19563	. . 3 ⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
15	13, 14	brab2a 5782	. 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 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  ℕ₀cn0 12524  ↑cexp 14099  ℙcprime 16705  Basecbs 17245  Grpcgrp 18964  odcod 19557   pGrp cpgp 19559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-iota 6516  df-fv 6571  df-ov 7434  df-pgp 19563

This theorem is referenced by:  pgpprm  19626  pgpgrp  19627  pgpfi1  19628  subgpgp  19630  pgpfi  19638