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Theorem ispgp 19460
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 486 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑔 = 𝐺)
21fveq2d 6896 . . . . 5 ((𝑝 = 𝑃𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
3 ispgp.1 . . . . 5 𝑋 = (Base‘𝐺)
42, 3eqtr4di 2791 . . . 4 ((𝑝 = 𝑃𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
51fveq2d 6896 . . . . . . . 8 ((𝑝 = 𝑃𝑔 = 𝐺) → (od‘𝑔) = (od‘𝐺))
6 ispgp.2 . . . . . . . 8 𝑂 = (od‘𝐺)
75, 6eqtr4di 2791 . . . . . . 7 ((𝑝 = 𝑃𝑔 = 𝐺) → (od‘𝑔) = 𝑂)
87fveq1d 6894 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → ((od‘𝑔)‘𝑥) = (𝑂𝑥))
9 simpl 484 . . . . . . 7 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑝 = 𝑃)
109oveq1d 7424 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑝𝑛) = (𝑃𝑛))
118, 10eqeq12d 2749 . . . . 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 3343 . . 3 ((𝑝 = 𝑃𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
14 df-pgp 19398 . . 3 pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛))}
1513, 14brab2a 5770 . 2 (𝑃 pGrp 𝐺 ↔ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
16 df-3an 1090 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071   class class class wbr 5149  cfv 6544  (class class class)co 7409  0cn0 12472  cexp 14027  cprime 16608  Basecbs 17144  Grpcgrp 18819  odcod 19392   pGrp cpgp 19394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-iota 6496  df-fv 6552  df-ov 7412  df-pgp 19398
This theorem is referenced by:  pgpprm  19461  pgpgrp  19462  pgpfi1  19463  subgpgp  19465  pgpfi  19473
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