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Theorem sylow1 18224
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 2771 . . 3 (+g𝐺) = (+g𝐺)
8 eqid 2771 . . 3 {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}
9 oveq2 6803 . . . . . . 7 (𝑠 = 𝑧 → (𝑢(+g𝐺)𝑠) = (𝑢(+g𝐺)𝑧))
109cbvmptv 4885 . . . . . 6 (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧))
11 oveq1 6802 . . . . . . 7 (𝑢 = 𝑥 → (𝑢(+g𝐺)𝑧) = (𝑥(+g𝐺)𝑧))
1211mpteq2dv 4880 . . . . . 6 (𝑢 = 𝑥 → (𝑧𝑣 ↦ (𝑢(+g𝐺)𝑧)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1310, 12syl5eq 2817 . . . . 5 (𝑢 = 𝑥 → (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
1413rneqd 5490 . . . 4 (𝑢 = 𝑥 → ran (𝑠𝑣 ↦ (𝑢(+g𝐺)𝑠)) = ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)))
15 mpteq1 4872 . . . . 5 (𝑣 = 𝑦 → (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1615rneqd 5490 . . . 4 (𝑣 = 𝑦 → ran (𝑧𝑣 ↦ (𝑥(+g𝐺)𝑧)) = ran (𝑧𝑦 ↦ (𝑥(+g𝐺)𝑧)))
1714, 16cbvmpt2v 6885 . . 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 4407 . . . . . 6 ((𝑎 = 𝑥𝑏 = 𝑦) → {𝑎, 𝑏} = {𝑥, 𝑦})
1918sseq1d 3781 . . . . 5 ((𝑎 = 𝑥𝑏 = 𝑦) → ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)} ↔ {𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}))
20 oveq2 6803 . . . . . . 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 2787 . . . . . 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 3200 . . . . 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 616 . . . 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 4857 . . 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 18221 . 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 466 . . 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 466 . . 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 466 . . 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 466 . . 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 466 . . 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 754 . . 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 2771 . . 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 756 . . 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 18223 . 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 3183 1 (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wrex 3062  {crab 3065  wss 3723  𝒫 cpw 4298  {cpr 4319   class class class wbr 4787  {copab 4847  cmpt 4864  ran crn 5251  cfv 6030  (class class class)co 6795  cmpt2 6797  [cec 7897  Fincfn 8112  cle 10280  cmin 10471  0cn0 11498  cexp 13066  chash 13320  cdvds 15188  cprime 15591   pCnt cpc 15747  Basecbs 16063  +gcplusg 16148  Grpcgrp 17629  SubGrpcsubg 17795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-inf2 8705  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218  ax-pre-sup 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-disj 4756  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-2o 7717  df-oadd 7720  df-er 7899  df-ec 7901  df-qs 7905  df-map 8014  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-sup 8507  df-inf 8508  df-oi 8574  df-card 8968  df-cda 9195  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-div 10890  df-nn 11226  df-2 11284  df-3 11285  df-n0 11499  df-xnn0 11570  df-z 11584  df-uz 11893  df-q 11996  df-rp 12035  df-fz 12533  df-fzo 12673  df-fl 12800  df-mod 12876  df-seq 13008  df-exp 13067  df-fac 13264  df-bc 13293  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-sum 14624  df-dvds 15189  df-gcd 15424  df-prm 15592  df-pc 15748  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-subg 17798  df-eqg 17800  df-ga 17929
This theorem is referenced by:  odcau  18225  slwhash  18245
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