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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 7167 | . . . . . 6 ⊢ (𝑠 = 𝑧 → (𝑢 + 𝑠) = (𝑢 + 𝑧)) | |
8 | 7 | cbvmptv 5172 | . . . . 5 ⊢ (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑢 + 𝑧)) |
9 | oveq1 7166 | . . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑢 + 𝑧) = (𝑥 + 𝑧)) | |
10 | 9 | mpteq2dv 5165 | . . . . 5 ⊢ (𝑢 = 𝑥 → (𝑧 ∈ 𝑣 ↦ (𝑢 + 𝑧)) = (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧))) |
11 | 8, 10 | syl5eq 2871 | . . . 4 ⊢ (𝑢 = 𝑥 → (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧))) |
12 | 11 | rneqd 5811 | . . 3 ⊢ (𝑢 = 𝑥 → ran (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠)) = ran (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧))) |
13 | mpteq1 5157 | . . . 4 ⊢ (𝑣 = 𝑦 → (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) | |
14 | 13 | rneqd 5811 | . . 3 ⊢ (𝑣 = 𝑦 → ran (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) |
15 | 12, 14 | cbvmpov 7252 | . 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 18750 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 ⊆ wss 3939 class class class wbr 5069 ↦ cmpt 5149 ran crn 5559 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 / cqs 8291 Fincfn 8512 ↑cexp 13432 ♯chash 13693 pCnt cpc 16176 Basecbs 16486 ↾s cress 16487 +gcplusg 16568 -gcsg 18108 SubGrpcsubg 18276 ~QG cqg 18278 pGrp cpgp 18657 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-disj 5035 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-omul 8110 df-er 8292 df-ec 8294 df-qs 8298 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-oi 8977 df-dju 9333 df-card 9371 df-acn 9374 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 df-dvds 15611 df-gcd 15847 df-prm 16019 df-pc 16177 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-eqg 18281 df-ga 18423 df-od 18659 df-pgp 18661 |
This theorem is referenced by: slwhash 18752 sylow2 18754 |
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