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Theorem sylow1 19520
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 2726 . . 3 (+g𝐺) = (+g𝐺)
8 eqid 2726 . . 3 {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}
9 oveq2 7412 . . . . . . 7 (𝑠 = 𝑧 → (𝑢(+g𝐺)𝑠) = (𝑢(+g𝐺)𝑧))
109cbvmptv 5254 . . . . . 6 (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧))
11 oveq1 7411 . . . . . . 7 (𝑢 = 𝑥 → (𝑢(+g𝐺)𝑧) = (𝑥(+g𝐺)𝑧))
1211mpteq2dv 5243 . . . . . 6 (𝑢 = 𝑥 → (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1310, 12eqtrid 2778 . . . . 5 (𝑢 = 𝑥 → (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1413rneqd 5930 . . . 4 (𝑢 = 𝑥 → ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
15 mpteq1 5234 . . . . 5 (𝑣 = 𝑦 → (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1615rneqd 5930 . . . 4 (𝑣 = 𝑦 → ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = ran (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1714, 16cbvmpov 7499 . . 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 4734 . . . . . 6 ((𝑎 = 𝑥𝑏 = 𝑦) → {𝑎, 𝑏} = {𝑥, 𝑦})
1918sseq1d 4008 . . . . 5 ((𝑎 = 𝑥𝑏 = 𝑦) → ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↔ {𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}))
20 oveq2 7412 . . . . . . 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 2740 . . . . . 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 3172 . . . . 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 630 . . . 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 5214 . . 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 19517 . 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 768 . . 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 2726 . . 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 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))) → (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 19519 . 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 3155 1 (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wrex 3064  {crab 3426  wss 3943  𝒫 cpw 4597  {cpr 4625   class class class wbr 5141  {copab 5203  cmpt 5224  ran crn 5670  cfv 6536  (class class class)co 7404  cmpo 7406  [cec 8700  Fincfn 8938  cle 11250  cmin 11445  0cn0 12473  cexp 14029  chash 14292  cdvds 16201  cprime 16612   pCnt cpc 16775  Basecbs 17150  +gcplusg 17203  Grpcgrp 18860  SubGrpcsubg 19044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-disj 5107  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-2o 8465  df-oadd 8468  df-er 8702  df-ec 8704  df-qs 8708  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-inf 9437  df-oi 9504  df-dju 9895  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-3 12277  df-n0 12474  df-xnn0 12546  df-z 12560  df-uz 12824  df-q 12934  df-rp 12978  df-fz 13488  df-fzo 13631  df-fl 13760  df-mod 13838  df-seq 13970  df-exp 14030  df-fac 14236  df-bc 14265  df-hash 14293  df-cj 15049  df-re 15050  df-im 15051  df-sqrt 15185  df-abs 15186  df-clim 15435  df-sum 15636  df-dvds 16202  df-gcd 16440  df-prm 16613  df-pc 16776  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-0g 17393  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-grp 18863  df-minusg 18864  df-subg 19047  df-eqg 19049  df-ga 19203
This theorem is referenced by:  odcau  19521  slwhash  19541
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