MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sylow1 Structured version   Visualization version   GIF version

Theorem sylow1 18728
Description: Sylow's first theorem. If 𝑃𝑁 is a prime power that divides the cardinality of 𝐺, then 𝐺 has a supgroup with size 𝑃𝑁. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypotheses
Ref Expression
sylow1.x 𝑋 = (Base‘𝐺)
sylow1.g (𝜑𝐺 ∈ Grp)
sylow1.f (𝜑𝑋 ∈ Fin)
sylow1.p (𝜑𝑃 ∈ ℙ)
sylow1.n (𝜑𝑁 ∈ ℕ0)
sylow1.d (𝜑 → (𝑃𝑁) ∥ (♯‘𝑋))
Assertion
Ref Expression
sylow1 (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃𝑁))
Distinct variable groups:   𝑔,𝑁   𝑔,𝑋   𝑔,𝐺   𝑃,𝑔   𝜑,𝑔

Proof of Theorem sylow1
Dummy variables 𝑎 𝑏 𝑠 𝑢 𝑥 𝑦 𝑧 𝑘 𝑡 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sylow1.x . . 3 𝑋 = (Base‘𝐺)
2 sylow1.g . . 3 (𝜑𝐺 ∈ Grp)
3 sylow1.f . . 3 (𝜑𝑋 ∈ Fin)
4 sylow1.p . . 3 (𝜑𝑃 ∈ ℙ)
5 sylow1.n . . 3 (𝜑𝑁 ∈ ℕ0)
6 sylow1.d . . 3 (𝜑 → (𝑃𝑁) ∥ (♯‘𝑋))
7 eqid 2824 . . 3 (+g𝐺) = (+g𝐺)
8 eqid 2824 . . 3 {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}
9 oveq2 7157 . . . . . . 7 (𝑠 = 𝑧 → (𝑢(+g𝐺)𝑠) = (𝑢(+g𝐺)𝑧))
109cbvmptv 5155 . . . . . 6 (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧))
11 oveq1 7156 . . . . . . 7 (𝑢 = 𝑥 → (𝑢(+g𝐺)𝑧) = (𝑥(+g𝐺)𝑧))
1211mpteq2dv 5148 . . . . . 6 (𝑢 = 𝑥 → (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1310, 12syl5eq 2871 . . . . 5 (𝑢 = 𝑥 → (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1413rneqd 5795 . . . 4 (𝑢 = 𝑥 → ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
15 mpteq1 5140 . . . . 5 (𝑣 = 𝑦 → (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1615rneqd 5795 . . . 4 (𝑣 = 𝑦 → ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = ran (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1714, 16cbvmpov 7242 . . 3 (𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠))) = (𝑥𝑋, 𝑦 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
18 preq12 4656 . . . . . 6 ((𝑎 = 𝑥𝑏 = 𝑦) → {𝑎, 𝑏} = {𝑥, 𝑦})
1918sseq1d 3984 . . . . 5 ((𝑎 = 𝑥𝑏 = 𝑦) → ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↔ {𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}))
20 oveq2 7157 . . . . . . 7 (𝑎 = 𝑥 → (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑥))
21 id 22 . . . . . . 7 (𝑏 = 𝑦𝑏 = 𝑦)
2220, 21eqeqan12d 2841 . . . . . 6 ((𝑎 = 𝑥𝑏 = 𝑦) → ((𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏 ↔ (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑥) = 𝑦))
2322rexbidv 3289 . . . . 5 ((𝑎 = 𝑥𝑏 = 𝑦) → (∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏 ↔ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑥) = 𝑦))
2419, 23anbi12d 633 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑦) → (({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏) ↔ ({𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑥) = 𝑦)))
2524cbvopabv 5124 . . 3 {⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑥) = 𝑦)}
261, 2, 3, 4, 5, 6, 7, 8, 17, 25sylow1lem3 18725 . 2 (𝜑 → ∃ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
272adantr 484 . . 3 ((𝜑 ∧ ( ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → 𝐺 ∈ Grp)
283adantr 484 . . 3 ((𝜑 ∧ ( ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → 𝑋 ∈ Fin)
294adantr 484 . . 3 ((𝜑 ∧ ( ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → 𝑃 ∈ ℙ)
305adantr 484 . . 3 ((𝜑 ∧ ( ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → 𝑁 ∈ ℕ0)
316adantr 484 . . 3 ((𝜑 ∧ ( ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → (𝑃𝑁) ∥ (♯‘𝑋))
32 simprl 770 . . 3 ((𝜑 ∧ ( ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)})
33 eqid 2824 . . 3 {𝑡𝑋 ∣ (𝑡(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))) = } = {𝑡𝑋 ∣ (𝑡(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))) = }
34 simprr 772 . . 3 ((𝜑 ∧ ( ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
351, 27, 28, 29, 30, 31, 7, 8, 17, 25, 32, 33, 34sylow1lem5 18727 . 2 ((𝜑 ∧ ( ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ (𝑃 pCnt (♯‘[]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ∧ ∃𝑘𝑋 (𝑘(𝑢𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↦ ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃𝑁))
3626, 35rexlimddv 3283 1 (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wrex 3134  {crab 3137  wss 3919  𝒫 cpw 4522  {cpr 4552   class class class wbr 5052  {copab 5114  cmpt 5132  ran crn 5543  cfv 6343  (class class class)co 7149  cmpo 7151  [cec 8283  Fincfn 8505  cle 10674  cmin 10868  0cn0 11894  cexp 13434  chash 13695  cdvds 15607  cprime 16013   pCnt cpc 16171  Basecbs 16483  +gcplusg 16565  Grpcgrp 18103  SubGrpcsubg 18273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-disj 5018  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-ec 8287  df-qs 8291  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-oi 8971  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-xnn0 11965  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-fz 12895  df-fzo 13038  df-fl 13166  df-mod 13242  df-seq 13374  df-exp 13435  df-fac 13639  df-bc 13668  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043  df-dvds 15608  df-gcd 15842  df-prm 16014  df-pc 16172  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-subg 18276  df-eqg 18278  df-ga 18420
This theorem is referenced by:  odcau  18729  slwhash  18749
  Copyright terms: Public domain W3C validator