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Mirrors > Home > MPE Home > Th. List > sylow2 | Structured version Visualization version GIF version |
Description: Sylow's second theorem. See also sylow2b 18238 for the "hard" part of the proof. Any two Sylow 𝑃-subgroups are conjugate to one another, and hence the same size, namely 𝑃↑(𝑃 pCnt ∣ 𝑋 ∣ ) (see fislw 18240). This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.) |
Ref | Expression |
---|---|
sylow2.x | ⊢ 𝑋 = (Base‘𝐺) |
sylow2.f | ⊢ (𝜑 → 𝑋 ∈ Fin) |
sylow2.h | ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺)) |
sylow2.k | ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) |
sylow2.a | ⊢ + = (+g‘𝐺) |
sylow2.d | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
sylow2 | ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylow2.f | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | 1 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝑋 ∈ Fin) |
3 | sylow2.k | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) | |
4 | slwsubg 18225 | . . . . . . . 8 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺)) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) |
6 | 5 | adantr 466 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐾 ∈ (SubGrp‘𝐺)) |
7 | simprl 754 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝑔 ∈ 𝑋) | |
8 | sylow2.x | . . . . . . 7 ⊢ 𝑋 = (Base‘𝐺) | |
9 | sylow2.a | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
10 | sylow2.d | . . . . . . 7 ⊢ − = (-g‘𝐺) | |
11 | eqid 2771 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) = (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) | |
12 | 8, 9, 10, 11 | conjsubg 17893 | . . . . . 6 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺)) |
13 | 6, 7, 12 | syl2anc 573 | . . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺)) |
14 | 8 | subgss 17796 | . . . . 5 ⊢ (ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ⊆ 𝑋) |
15 | 13, 14 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ⊆ 𝑋) |
16 | ssfi 8334 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ⊆ 𝑋) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ Fin) | |
17 | 2, 15, 16 | syl2anc 573 | . . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ Fin) |
18 | simprr 756 | . . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | |
19 | sylow2.h | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺)) | |
20 | 8, 1, 19 | slwhash 18239 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
21 | 8, 1, 3 | slwhash 18239 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
22 | 20, 21 | eqtr4d 2808 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝐾)) |
23 | slwsubg 18225 | . . . . . . . . . 10 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺)) | |
24 | 19, 23 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) |
25 | 8 | subgss 17796 | . . . . . . . . 9 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋) |
26 | 24, 25 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ 𝑋) |
27 | ssfi 8334 | . . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝐻 ⊆ 𝑋) → 𝐻 ∈ Fin) | |
28 | 1, 26, 27 | syl2anc 573 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Fin) |
29 | 8 | subgss 17796 | . . . . . . . . 9 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋) |
30 | 5, 29 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ⊆ 𝑋) |
31 | ssfi 8334 | . . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝐾 ⊆ 𝑋) → 𝐾 ∈ Fin) | |
32 | 1, 30, 31 | syl2anc 573 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Fin) |
33 | hashen 13332 | . . . . . . 7 ⊢ ((𝐻 ∈ Fin ∧ 𝐾 ∈ Fin) → ((♯‘𝐻) = (♯‘𝐾) ↔ 𝐻 ≈ 𝐾)) | |
34 | 28, 32, 33 | syl2anc 573 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐻) = (♯‘𝐾) ↔ 𝐻 ≈ 𝐾)) |
35 | 22, 34 | mpbid 222 | . . . . 5 ⊢ (𝜑 → 𝐻 ≈ 𝐾) |
36 | 35 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 ≈ 𝐾) |
37 | 8, 9, 10, 11 | conjsubgen 17894 | . . . . 5 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
38 | 6, 7, 37 | syl2anc 573 | . . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
39 | entr 8159 | . . . 4 ⊢ ((𝐻 ≈ 𝐾 ∧ 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | |
40 | 36, 38, 39 | syl2anc 573 | . . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
41 | fisseneq 8325 | . . 3 ⊢ ((ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ Fin ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∧ 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) → 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | |
42 | 17, 18, 40, 41 | syl3anc 1476 | . 2 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
43 | eqid 2771 | . . . . 5 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻) | |
44 | 43 | slwpgp 18228 | . . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾s 𝐻)) |
45 | 19, 44 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻)) |
46 | 8, 1, 24, 5, 9, 45, 21, 10 | sylow2b 18238 | . 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
47 | 42, 46 | reximddv 3166 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∃wrex 3062 ⊆ wss 3723 class class class wbr 4786 ↦ cmpt 4863 ran crn 5250 ‘cfv 6029 (class class class)co 6791 ≈ cen 8104 Fincfn 8107 ↑cexp 13060 ♯chash 13314 pCnt cpc 15741 Basecbs 16057 ↾s cress 16058 +gcplusg 16142 -gcsg 17625 SubGrpcsubg 17789 pGrp cpgp 18146 pSyl cslw 18147 |
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 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 |
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 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-disj 4755 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-2o 7712 df-oadd 7715 df-omul 7716 df-er 7894 df-ec 7896 df-qs 7900 df-map 8009 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-sup 8502 df-inf 8503 df-oi 8569 df-card 8963 df-acn 8966 df-cda 9190 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-n0 11493 df-xnn0 11564 df-z 11578 df-uz 11887 df-q 11990 df-rp 12029 df-fz 12527 df-fzo 12667 df-fl 12794 df-mod 12870 df-seq 13002 df-exp 13061 df-fac 13258 df-bc 13287 df-hash 13315 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-clim 14420 df-sum 14618 df-dvds 15183 df-gcd 15418 df-prm 15586 df-pc 15742 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16155 df-0g 16303 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-mulg 17742 df-subg 17792 df-eqg 17794 df-ghm 17859 df-ga 17923 df-od 18148 df-pgp 18150 df-slw 18151 |
This theorem is referenced by: sylow3lem3 18244 sylow3lem6 18247 |
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