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Theorem sylow1 18402
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 2778 . . 3 (+g𝐺) = (+g𝐺)
8 eqid 2778 . . 3 {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}
9 oveq2 6930 . . . . . . 7 (𝑠 = 𝑧 → (𝑢(+g𝐺)𝑠) = (𝑢(+g𝐺)𝑧))
109cbvmptv 4985 . . . . . 6 (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧))
11 oveq1 6929 . . . . . . 7 (𝑢 = 𝑥 → (𝑢(+g𝐺)𝑧) = (𝑥(+g𝐺)𝑧))
1211mpteq2dv 4980 . . . . . 6 (𝑢 = 𝑥 → (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1310, 12syl5eq 2826 . . . . 5 (𝑢 = 𝑥 → (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1413rneqd 5598 . . . 4 (𝑢 = 𝑥 → ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
15 mpteq1 4972 . . . . 5 (𝑣 = 𝑦 → (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1615rneqd 5598 . . . 4 (𝑣 = 𝑦 → ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = ran (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1714, 16cbvmpt2v 7012 . . 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 4502 . . . . . 6 ((𝑎 = 𝑥𝑏 = 𝑦) → {𝑎, 𝑏} = {𝑥, 𝑦})
1918sseq1d 3851 . . . . 5 ((𝑎 = 𝑥𝑏 = 𝑦) → ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↔ {𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}))
20 oveq2 6930 . . . . . . 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 2794 . . . . . 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 3237 . . . . 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 624 . . . 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 4958 . . 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 18399 . 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 474 . . 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 474 . . 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 474 . . 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 474 . . 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 474 . . 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 761 . . 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 2778 . . 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 763 . . 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 18401 . 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 3218 1 (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wrex 3091  {crab 3094  wss 3792  𝒫 cpw 4379  {cpr 4400   class class class wbr 4886  {copab 4948  cmpt 4965  ran crn 5356  cfv 6135  (class class class)co 6922  cmpt2 6924  [cec 8024  Fincfn 8241  cle 10412  cmin 10606  0cn0 11642  cexp 13178  chash 13435  cdvds 15387  cprime 15790   pCnt cpc 15945  Basecbs 16255  +gcplusg 16338  Grpcgrp 17809  SubGrpcsubg 17972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-disj 4855  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-ec 8028  df-qs 8032  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-q 12096  df-rp 12138  df-fz 12644  df-fzo 12785  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179  df-fac 13379  df-bc 13408  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825  df-dvds 15388  df-gcd 15623  df-prm 15791  df-pc 15946  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813  df-subg 17975  df-eqg 17977  df-ga 18106
This theorem is referenced by:  odcau  18403  slwhash  18423
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