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Mirrors > Home > MPE Home > Th. List > sylow2b | Structured version Visualization version GIF version |
Description: Sylow's second theorem. Any 𝑃-group 𝐻 is a subgroup of a conjugated 𝑃-group 𝐾 of order 𝑃↑𝑛 ∥ (♯‘𝑋) with 𝑛 maximal. This is usually stated under the assumption that 𝐾 is a Sylow subgroup, but we use a slightly different definition, whose equivalence to this one requires this theorem. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.) |
Ref | Expression |
---|---|
sylow2b.x | ⊢ 𝑋 = (Base‘𝐺) |
sylow2b.xf | ⊢ (𝜑 → 𝑋 ∈ Fin) |
sylow2b.h | ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) |
sylow2b.k | ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) |
sylow2b.a | ⊢ + = (+g‘𝐺) |
sylow2b.hp | ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻)) |
sylow2b.kn | ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
sylow2b.d | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
sylow2b | ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylow2b.x | . 2 ⊢ 𝑋 = (Base‘𝐺) | |
2 | sylow2b.xf | . 2 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
3 | sylow2b.h | . 2 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) | |
4 | sylow2b.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) | |
5 | sylow2b.a | . 2 ⊢ + = (+g‘𝐺) | |
6 | eqid 2824 | . 2 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾) | |
7 | oveq2 6912 | . . . . . 6 ⊢ (𝑠 = 𝑧 → (𝑢 + 𝑠) = (𝑢 + 𝑧)) | |
8 | 7 | cbvmptv 4972 | . . . . 5 ⊢ (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑢 + 𝑧)) |
9 | oveq1 6911 | . . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑢 + 𝑧) = (𝑥 + 𝑧)) | |
10 | 9 | mpteq2dv 4967 | . . . . 5 ⊢ (𝑢 = 𝑥 → (𝑧 ∈ 𝑣 ↦ (𝑢 + 𝑧)) = (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧))) |
11 | 8, 10 | syl5eq 2872 | . . . 4 ⊢ (𝑢 = 𝑥 → (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧))) |
12 | 11 | rneqd 5584 | . . 3 ⊢ (𝑢 = 𝑥 → ran (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠)) = ran (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧))) |
13 | mpteq1 4959 | . . . 4 ⊢ (𝑣 = 𝑦 → (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) | |
14 | 13 | rneqd 5584 | . . 3 ⊢ (𝑣 = 𝑦 → ran (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) |
15 | 12, 14 | cbvmpt2v 6994 | . 2 ⊢ (𝑢 ∈ 𝐻, 𝑣 ∈ (𝑋 / (𝐺 ~QG 𝐾)) ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠))) = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / (𝐺 ~QG 𝐾)) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) |
16 | sylow2b.hp | . 2 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻)) | |
17 | sylow2b.kn | . 2 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) | |
18 | sylow2b.d | . 2 ⊢ − = (-g‘𝐺) | |
19 | 1, 2, 3, 4, 5, 6, 15, 16, 17, 18 | sylow2blem3 18387 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∃wrex 3117 ⊆ wss 3797 class class class wbr 4872 ↦ cmpt 4951 ran crn 5342 ‘cfv 6122 (class class class)co 6904 ↦ cmpt2 6906 / cqs 8007 Fincfn 8221 ↑cexp 13153 ♯chash 13409 pCnt cpc 15911 Basecbs 16221 ↾s cress 16222 +gcplusg 16304 -gcsg 17777 SubGrpcsubg 17938 ~QG cqg 17940 pGrp cpgp 18296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-disj 4841 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-omul 7830 df-er 8008 df-ec 8010 df-qs 8014 df-map 8123 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-acn 9080 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-xnn0 11690 df-z 11704 df-uz 11968 df-q 12071 df-rp 12112 df-fz 12619 df-fzo 12760 df-fl 12887 df-mod 12963 df-seq 13095 df-exp 13154 df-fac 13353 df-bc 13382 df-hash 13410 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-clim 14595 df-sum 14793 df-dvds 15357 df-gcd 15589 df-prm 15757 df-pc 15912 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-0g 16454 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-grp 17778 df-minusg 17779 df-sbg 17780 df-mulg 17894 df-subg 17941 df-eqg 17943 df-ga 18072 df-od 18298 df-pgp 18300 |
This theorem is referenced by: slwhash 18389 sylow2 18391 |
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