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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 19534 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 19536). 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 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝑋 ∈ Fin) |
3 | sylow2.k | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) | |
4 | slwsubg 19521 | . . . . . . 7 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) |
6 | simprl 767 | . . . . . 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 2730 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) = (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) | |
11 | 7, 8, 9, 10 | conjsubg 19166 | . . . . . 6 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺)) |
12 | 5, 6, 11 | syl2an2r 681 | . . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ (SubGrp‘𝐺)) |
13 | 7 | subgss 19045 | . . . . 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 9271 | . . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ∈ Fin) |
16 | simprr 769 | . . 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 19535 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
19 | 7, 1, 3 | slwhash 19535 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
20 | 18, 19 | eqtr4d 2773 | . . . . 5 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝐾)) |
21 | slwsubg 19521 | . . . . . . . . 9 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺)) | |
22 | 17, 21 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) |
23 | 7 | subgss 19045 | . . . . . . . 8 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋) |
24 | 22, 23 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ 𝑋) |
25 | 1, 24 | ssfid 9271 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Fin) |
26 | 7 | subgss 19045 | . . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋) |
27 | 5, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋) |
28 | 1, 27 | ssfid 9271 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin) |
29 | hashen 14313 | . . . . . 6 ⊢ ((𝐻 ∈ Fin ∧ 𝐾 ∈ Fin) → ((♯‘𝐻) = (♯‘𝐾) ↔ 𝐻 ≈ 𝐾)) | |
30 | 25, 28, 29 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → ((♯‘𝐻) = (♯‘𝐾) ↔ 𝐻 ≈ 𝐾)) |
31 | 20, 30 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝐻 ≈ 𝐾) |
32 | 7, 8, 9, 10 | conjsubgen 19167 | . . . . 5 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
33 | 5, 6, 32 | syl2an2r 681 | . . . 4 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
34 | entr 9006 | . . . 4 ⊢ ((𝐻 ≈ 𝐾 ∧ 𝐾 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | |
35 | 31, 33, 34 | syl2an2r 681 | . . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 ≈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
36 | fisseneq 9261 | . . 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 1369 | . 2 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))) → 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
38 | eqid 2730 | . . . . 5 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻) | |
39 | 38 | slwpgp 19524 | . . . 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 19534 | . 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
42 | 37, 41 | reximddv 3169 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∃wrex 3068 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ran crn 5678 ‘cfv 6544 (class class class)co 7413 ≈ cen 8940 Fincfn 8943 ↑cexp 14033 ♯chash 14296 pCnt cpc 16775 Basecbs 17150 ↾s cress 17179 +gcplusg 17203 -gcsg 18859 SubGrpcsubg 19038 pGrp cpgp 19437 pSyl cslw 19438 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-omul 8475 df-er 8707 df-ec 8709 df-qs 8713 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-acn 9941 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-n0 12479 df-xnn0 12551 df-z 12565 df-uz 12829 df-q 12939 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14034 df-fac 14240 df-bc 14269 df-hash 14297 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-dvds 16204 df-gcd 16442 df-prm 16615 df-pc 16776 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-0g 17393 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18708 df-grp 18860 df-minusg 18861 df-sbg 18862 df-mulg 18989 df-subg 19041 df-eqg 19043 df-ghm 19130 df-ga 19197 df-od 19439 df-pgp 19441 df-slw 19442 |
This theorem is referenced by: sylow3lem3 19540 sylow3lem6 19543 |
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