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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 19491 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 19493). 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 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝑋 ∈ Fin) |
3 | sylow2.k | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) | |
4 | slwsubg 19478 | . . . . . . 7 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) |
6 | simprl 770 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝑔 ∈ 𝑋) | |
7 | sylow2.x | . . . . . . 7 ⊢ 𝑋 = (Base‘𝐺) | |
8 | sylow2.a | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
9 | sylow2.d | . . . . . . 7 ⊢ − = (-g‘𝐺) | |
10 | eqid 2733 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) = (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) | |
11 | 7, 8, 9, 10 | conjsubg 19124 | . . . . . 6 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺)) |
12 | 5, 6, 11 | syl2an2r 684 | . . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺)) |
13 | 7 | subgss 19007 | . . . . 5 ⊢ (ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ⊆ 𝑋) |
14 | 12, 13 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ⊆ 𝑋) |
15 | 2, 14 | ssfid 9267 | . . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ Fin) |
16 | simprr 772 | . . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | |
17 | sylow2.h | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺)) | |
18 | 7, 1, 17 | slwhash 19492 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
19 | 7, 1, 3 | slwhash 19492 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
20 | 18, 19 | eqtr4d 2776 | . . . . 5 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝐾)) |
21 | slwsubg 19478 | . . . . . . . . 9 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺)) | |
22 | 17, 21 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) |
23 | 7 | subgss 19007 | . . . . . . . 8 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋) |
24 | 22, 23 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ 𝑋) |
25 | 1, 24 | ssfid 9267 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Fin) |
26 | 7 | subgss 19007 | . . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋) |
27 | 5, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋) |
28 | 1, 27 | ssfid 9267 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin) |
29 | hashen 14307 | . . . . . 6 ⊢ ((𝐻 ∈ Fin ∧ 𝐾 ∈ Fin) → ((♯‘𝐻) = (♯‘𝐾) ↔ 𝐻 ≈ 𝐾)) | |
30 | 25, 28, 29 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → ((♯‘𝐻) = (♯‘𝐾) ↔ 𝐻 ≈ 𝐾)) |
31 | 20, 30 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝐻 ≈ 𝐾) |
32 | 7, 8, 9, 10 | conjsubgen 19125 | . . . . 5 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
33 | 5, 6, 32 | syl2an2r 684 | . . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
34 | entr 9002 | . . . 4 ⊢ ((𝐻 ≈ 𝐾 ∧ 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | |
35 | 31, 33, 34 | syl2an2r 684 | . . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
36 | fisseneq 9257 | . . 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))) | |
37 | 15, 16, 35, 36 | syl3anc 1372 | . 2 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
38 | eqid 2733 | . . . . 5 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻) | |
39 | 38 | slwpgp 19481 | . . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾s 𝐻)) |
40 | 17, 39 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻)) |
41 | 7, 1, 22, 5, 8, 40, 19, 9 | sylow2b 19491 | . 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
42 | 37, 41 | reximddv 3172 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ran crn 5678 ‘cfv 6544 (class class class)co 7409 ≈ cen 8936 Fincfn 8939 ↑cexp 14027 ♯chash 14290 pCnt cpc 16769 Basecbs 17144 ↾s cress 17173 +gcplusg 17197 -gcsg 18821 SubGrpcsubg 19000 pGrp cpgp 19394 pSyl cslw 19395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-ec 8705 df-qs 8709 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-dvds 16198 df-gcd 16436 df-prm 16609 df-pc 16770 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-eqg 19005 df-ghm 19090 df-ga 19154 df-od 19396 df-pgp 19398 df-slw 19399 |
This theorem is referenced by: sylow3lem3 19497 sylow3lem6 19500 |
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