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Mirrors > Home > MPE Home > Th. List > orbsta2 | Structured version Visualization version GIF version |
Description: Relation between the size of the orbit and the size of the stabilizer of a point in a finite group action. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Ref | Expression |
---|---|
orbsta2.x | ⊢ 𝑋 = (Base‘𝐺) |
orbsta2.h | ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴} |
orbsta2.r | ⊢ ∼ = (𝐺 ~QG 𝐻) |
orbsta2.o | ⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} |
Ref | Expression |
---|---|
orbsta2 | ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝐴]𝑂) · (♯‘𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orbsta2.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
2 | orbsta2.r | . . 3 ⊢ ∼ = (𝐺 ~QG 𝐻) | |
3 | orbsta2.h | . . . . 5 ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴} | |
4 | 1, 3 | gastacl 18439 | . . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺)) |
5 | 4 | adantr 483 | . . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝐻 ∈ (SubGrp‘𝐺)) |
6 | simpr 487 | . . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin) | |
7 | 1, 2, 5, 6 | lagsubg2 18341 | . 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝐻))) |
8 | pwfi 8819 | . . . . . 6 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin) | |
9 | 6, 8 | sylib 220 | . . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝒫 𝑋 ∈ Fin) |
10 | 1, 2 | eqger 18330 | . . . . . . 7 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋) |
11 | 5, 10 | syl 17 | . . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ∼ Er 𝑋) |
12 | 11 | qsss 8358 | . . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ⊆ 𝒫 𝑋) |
13 | 9, 12 | ssfid 8741 | . . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ∈ Fin) |
14 | eqid 2821 | . . . . . 6 ⊢ ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉) = ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉) | |
15 | orbsta2.o | . . . . . 6 ⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} | |
16 | 1, 3, 2, 14, 15 | orbsta 18443 | . . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂) |
17 | 16 | adantr 483 | . . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂) |
18 | 13, 17 | hasheqf1od 13715 | . . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘(𝑋 / ∼ )) = (♯‘[𝐴]𝑂)) |
19 | 18 | oveq1d 7171 | . 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ((♯‘(𝑋 / ∼ )) · (♯‘𝐻)) = ((♯‘[𝐴]𝑂) · (♯‘𝐻))) |
20 | 7, 19 | eqtrd 2856 | 1 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝐴]𝑂) · (♯‘𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 {crab 3142 ⊆ wss 3936 𝒫 cpw 4539 {cpr 4569 〈cop 4573 {copab 5128 ↦ cmpt 5146 ran crn 5556 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 Er wer 8286 [cec 8287 / cqs 8288 Fincfn 8509 · cmul 10542 ♯chash 13691 Basecbs 16483 SubGrpcsubg 18273 ~QG cqg 18275 GrpAct cga 18419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-ec 8291 df-qs 8295 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-subg 18276 df-eqg 18278 df-ga 18420 |
This theorem is referenced by: sylow1lem5 18727 sylow2alem2 18743 sylow3lem3 18754 |
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