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Theorem ispgp 17830
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 ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
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.
StepHypRef Expression
1 simpr 476 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑔 = 𝐺)
21fveq2d 6107 . . . . 5 ((𝑝 = 𝑃𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
3 ispgp.1 . . . . 5 𝑋 = (Base‘𝐺)
42, 3syl6eqr 2662 . . . 4 ((𝑝 = 𝑃𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
51fveq2d 6107 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (od‘𝑔) = (od‘𝐺))
6 ispgp.2 . . . . . . . 8 𝑂 = (od‘𝐺)
75, 6syl6eqr 2662 . . . . . . 7 ((𝑝 = 𝑃𝑔 = 𝐺) → (od‘𝑔) = 𝑂)
87fveq1d 6105 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → ((od‘𝑔)‘𝑥) = (𝑂𝑥))
9 simpl 472 . . . . . . 7 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑝 = 𝑃)
109oveq1d 6564 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑝𝑛) = (𝑃𝑛))
118, 10eqeq12d 2625 . . . . 5 ((𝑝 = 𝑃𝑔 = 𝐺) → (((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ (𝑂𝑥) = (𝑃𝑛)))
1211rexbidv 3034 . . . 4 ((𝑝 = 𝑃𝑔 = 𝐺) → (∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
134, 12raleqbidv 3129 . . 3 ((𝑝 = 𝑃𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
14 df-pgp 17773 . . 3 pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛))}
1513, 14brab2ga 5117 . 2 (𝑃 pGrp 𝐺 ↔ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
16 df-3an 1033 . 2 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)) ↔ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
1715, 16bitr4i 266 1 (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897   class class class wbr 4583  cfv 5804  (class class class)co 6549  0cn0 11169  cexp 12722  cprime 15223  Basecbs 15695  Grpcgrp 17245  odcod 17767   pGrp cpgp 17769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-iota 5768  df-fv 5812  df-ov 6552  df-pgp 17773
This theorem is referenced by:  pgpprm  17831  pgpgrp  17832  pgpfi1  17833  subgpgp  17835  pgpfi  17843
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