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Theorem ispgp 19610
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 484 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑔 = 𝐺)
21fveq2d 6910 . . . . 5 ((𝑝 = 𝑃𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
3 ispgp.1 . . . . 5 𝑋 = (Base‘𝐺)
42, 3eqtr4di 2795 . . . 4 ((𝑝 = 𝑃𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
51fveq2d 6910 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (od‘𝑔) = (od‘𝐺))
6 ispgp.2 . . . . . . . 8 𝑂 = (od‘𝐺)
75, 6eqtr4di 2795 . . . . . . 7 ((𝑝 = 𝑃𝑔 = 𝐺) → (od‘𝑔) = 𝑂)
87fveq1d 6908 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → ((od‘𝑔)‘𝑥) = (𝑂𝑥))
9 simpl 482 . . . . . . 7 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑝 = 𝑃)
109oveq1d 7446 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑝𝑛) = (𝑃𝑛))
118, 10eqeq12d 2753 . . . . 5 ((𝑝 = 𝑃𝑔 = 𝐺) → (((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ (𝑂𝑥) = (𝑃𝑛)))
1211rexbidv 3179 . . . 4 ((𝑝 = 𝑃𝑔 = 𝐺) → (∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
134, 12raleqbidv 3346 . . 3 ((𝑝 = 𝑃𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
14 df-pgp 19548 . . 3 pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛))}
1513, 14brab2a 5779 . 2 (𝑃 pGrp 𝐺 ↔ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
16 df-3an 1089 . 2 ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)) ↔ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
1715, 16bitr4i 278 1 (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070   class class class wbr 5143  cfv 6561  (class class class)co 7431  0cn0 12526  cexp 14102  cprime 16708  Basecbs 17247  Grpcgrp 18951  odcod 19542   pGrp cpgp 19544
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-iota 6514  df-fv 6569  df-ov 7434  df-pgp 19548
This theorem is referenced by:  pgpprm  19611  pgpgrp  19612  pgpfi1  19613  subgpgp  19615  pgpfi  19623
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