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Theorem ispgp 19379
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 6847 . . . . 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 6847 . . . . . . . 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 6845 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → ((od‘𝑔)‘𝑥) = (𝑂𝑥))
9 simpl 484 . . . . . . 7 ((𝑝 = 𝑃𝑔 = 𝐺) → 𝑝 = 𝑃)
109oveq1d 7373 . . . . . 6 ((𝑝 = 𝑃𝑔 = 𝐺) → (𝑝𝑛) = (𝑃𝑛))
118, 10eqeq12d 2749 . . . . 5 ((𝑝 = 𝑃𝑔 = 𝐺) → (((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ (𝑂𝑥) = (𝑃𝑛)))
1211rexbidv 3172 . . . 4 ((𝑝 = 𝑃𝑔 = 𝐺) → (∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
134, 12raleqbidv 3318 . . 3 ((𝑝 = 𝑃𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛) ↔ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))
14 df-pgp 19317 . . 3 pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛))}
1513, 14brab2a 5726 . 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 3061  wrex 3070   class class class wbr 5106  cfv 6497  (class class class)co 7358  0cn0 12418  cexp 13973  cprime 16552  Basecbs 17088  Grpcgrp 18753  odcod 19311   pGrp cpgp 19313
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 5257  ax-nul 5264  ax-pr 5385
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 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-iota 6449  df-fv 6505  df-ov 7361  df-pgp 19317
This theorem is referenced by:  pgpprm  19380  pgpgrp  19381  pgpfi1  19382  subgpgp  19384  pgpfi  19392
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