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

Theorem sylow1 19621
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 2737 . . 3 (+g𝐺) = (+g𝐺)
8 eqid 2737 . . 3 {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}
9 oveq2 7439 . . . . . . 7 (𝑠 = 𝑧 → (𝑢(+g𝐺)𝑠) = (𝑢(+g𝐺)𝑧))
109cbvmptv 5255 . . . . . 6 (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧))
11 oveq1 7438 . . . . . . 7 (𝑢 = 𝑥 → (𝑢(+g𝐺)𝑧) = (𝑥(+g𝐺)𝑧))
1211mpteq2dv 5244 . . . . . 6 (𝑢 = 𝑥 → (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1310, 12eqtrid 2789 . . . . 5 (𝑢 = 𝑥 → (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1413rneqd 5949 . . . 4 (𝑢 = 𝑥 → ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
15 mpteq1 5235 . . . . 5 (𝑣 = 𝑦 → (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1615rneqd 5949 . . . 4 (𝑣 = 𝑦 → ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = ran (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1714, 16cbvmpov 7528 . . 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 4735 . . . . . 6 ((𝑎 = 𝑥𝑏 = 𝑦) → {𝑎, 𝑏} = {𝑥, 𝑦})
1918sseq1d 4015 . . . . 5 ((𝑎 = 𝑥𝑏 = 𝑦) → ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↔ {𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}))
20 oveq2 7439 . . . . . . 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 2751 . . . . . 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 3179 . . . . 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 632 . . . 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 5216 . . 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 19618 . 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 480 . . 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 480 . . 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 480 . . 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 480 . . 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 480 . . 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 771 . . 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 2737 . . 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 773 . . 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 19620 . 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 3161 1 (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  wss 3951  𝒫 cpw 4600  {cpr 4628   class class class wbr 5143  {copab 5205  cmpt 5225  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  [cec 8743  Fincfn 8985  cle 11296  cmin 11492  0cn0 12526  cexp 14102  chash 14369  cdvds 16290  cprime 16708   pCnt cpc 16874  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  SubGrpcsubg 19138
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-subg 19141  df-eqg 19143  df-ga 19308
This theorem is referenced by:  odcau  19622  slwhash  19642
  Copyright terms: Public domain W3C validator