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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 2740 | . 2 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾) | |
7 | oveq2 7279 | . . . . . 6 ⊢ (𝑠 = 𝑧 → (𝑢 + 𝑠) = (𝑢 + 𝑧)) | |
8 | 7 | cbvmptv 5192 | . . . . 5 ⊢ (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑢 + 𝑧)) |
9 | oveq1 7278 | . . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑢 + 𝑧) = (𝑥 + 𝑧)) | |
10 | 9 | mpteq2dv 5181 | . . . . 5 ⊢ (𝑢 = 𝑥 → (𝑧 ∈ 𝑣 ↦ (𝑢 + 𝑧)) = (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧))) |
11 | 8, 10 | eqtrid 2792 | . . . 4 ⊢ (𝑢 = 𝑥 → (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧))) |
12 | 11 | rneqd 5846 | . . 3 ⊢ (𝑢 = 𝑥 → ran (𝑠 ∈ 𝑣 ↦ (𝑢 + 𝑠)) = ran (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧))) |
13 | mpteq1 5172 | . . . 4 ⊢ (𝑣 = 𝑦 → (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) | |
14 | 13 | rneqd 5846 | . . 3 ⊢ (𝑣 = 𝑦 → ran (𝑧 ∈ 𝑣 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) |
15 | 12, 14 | cbvmpov 7364 | . 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 19225 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 ⊆ wss 3892 class class class wbr 5079 ↦ cmpt 5162 ran crn 5591 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 / cqs 8480 Fincfn 8716 ↑cexp 13780 ♯chash 14042 pCnt cpc 16535 Basecbs 16910 ↾s cress 16939 +gcplusg 16960 -gcsg 18577 SubGrpcsubg 18747 ~QG cqg 18749 pGrp cpgp 19132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-disj 5045 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-oadd 8292 df-omul 8293 df-er 8481 df-ec 8483 df-qs 8487 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-inf 9180 df-oi 9247 df-dju 9660 df-card 9698 df-acn 9701 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12582 df-q 12688 df-rp 12730 df-fz 13239 df-fzo 13382 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-fac 13986 df-bc 14015 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-sum 15396 df-dvds 15962 df-gcd 16200 df-prm 16375 df-pc 16536 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-eqg 18752 df-ga 18894 df-od 19134 df-pgp 19136 |
This theorem is referenced by: slwhash 19227 sylow2 19229 |
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