Theorem ispgp 19565

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 485	. . . . . 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 2793	. . . 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 2793	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (od‘𝑔) = 𝑂)
8	7	fveq1d 6836	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → ((od‘𝑔)‘𝑥) = (𝑂‘𝑥))
9		simpl 483	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑝 = 𝑃)
10	9	oveq1d 7378	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑝↑𝑛) = (𝑃↑𝑛))
11	8, 10	eqeq12d 2756	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (((od‘𝑔)‘𝑥) = (𝑝↑𝑛) ↔ (𝑂‘𝑥) = (𝑃↑𝑛)))
12	11	rexbidv 3164	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
13	4, 12	raleqbidv 3314	. . 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 19503	. . 3 ⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
15	13, 14	brab2a 5718	. 2 ⊢ (𝑃 pGrp 𝐺 ↔ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
16		df-3an 1094	. 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 279	1 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = 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:  pgpprm  19566  pgpgrp  19567  pgpfi1  19568  subgpgp  19570  pgpfi  19578