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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 17565 | . . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺)) |
5 | 4 | adantr 480 | . . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝐻 ∈ (SubGrp‘𝐺)) |
6 | simpr 476 | . . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin) | |
7 | 1, 2, 5, 6 | lagsubg2 17478 | . 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘(𝑋 / ∼ )) · (#‘𝐻))) |
8 | eqid 2610 | . . . . . . 7 ⊢ ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉) = ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉) | |
9 | orbsta2.o | . . . . . . 7 ⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} | |
10 | 1, 3, 2, 8, 9 | orbsta 17569 | . . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂) |
11 | 10 | adantr 480 | . . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂) |
12 | fvex 6113 | . . . . . . . 8 ⊢ (Base‘𝐺) ∈ V | |
13 | 1, 12 | eqeltri 2684 | . . . . . . 7 ⊢ 𝑋 ∈ V |
14 | 13 | qsex 7693 | . . . . . 6 ⊢ (𝑋 / ∼ ) ∈ V |
15 | 14 | f1oen 7862 | . . . . 5 ⊢ (ran (𝑘 ∈ 𝑋 ↦ 〈[𝑘] ∼ , (𝑘 ⊕ 𝐴)〉):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂 → (𝑋 / ∼ ) ≈ [𝐴]𝑂) |
16 | 11, 15 | syl 17 | . . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ≈ [𝐴]𝑂) |
17 | pwfi 8144 | . . . . . . 7 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin) | |
18 | 6, 17 | sylib 207 | . . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝒫 𝑋 ∈ Fin) |
19 | 1, 2 | eqger 17467 | . . . . . . . 8 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋) |
20 | 5, 19 | syl 17 | . . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ∼ Er 𝑋) |
21 | 20 | qsss 7695 | . . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ⊆ 𝒫 𝑋) |
22 | ssfi 8065 | . . . . . 6 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑋 / ∼ ) ⊆ 𝒫 𝑋) → (𝑋 / ∼ ) ∈ Fin) | |
23 | 18, 21, 22 | syl2anc 691 | . . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ∈ Fin) |
24 | 16 | ensymd 7893 | . . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → [𝐴]𝑂 ≈ (𝑋 / ∼ )) |
25 | enfii 8062 | . . . . . 6 ⊢ (((𝑋 / ∼ ) ∈ Fin ∧ [𝐴]𝑂 ≈ (𝑋 / ∼ )) → [𝐴]𝑂 ∈ Fin) | |
26 | 23, 24, 25 | syl2anc 691 | . . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → [𝐴]𝑂 ∈ Fin) |
27 | hashen 12997 | . . . . 5 ⊢ (((𝑋 / ∼ ) ∈ Fin ∧ [𝐴]𝑂 ∈ Fin) → ((#‘(𝑋 / ∼ )) = (#‘[𝐴]𝑂) ↔ (𝑋 / ∼ ) ≈ [𝐴]𝑂)) | |
28 | 23, 26, 27 | syl2anc 691 | . . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ((#‘(𝑋 / ∼ )) = (#‘[𝐴]𝑂) ↔ (𝑋 / ∼ ) ≈ [𝐴]𝑂)) |
29 | 16, 28 | mpbird 246 | . . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (#‘(𝑋 / ∼ )) = (#‘[𝐴]𝑂)) |
30 | 29 | oveq1d 6564 | . 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ((#‘(𝑋 / ∼ )) · (#‘𝐻)) = ((#‘[𝐴]𝑂) · (#‘𝐻))) |
31 | 7, 30 | eqtrd 2644 | 1 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘[𝐴]𝑂) · (#‘𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 {cpr 4127 〈cop 4131 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 ran crn 5039 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Er wer 7626 [cec 7627 / cqs 7628 ≈ cen 7838 Fincfn 7841 · cmul 9820 #chash 12979 Basecbs 15695 SubGrpcsubg 17411 ~QG cqg 17413 GrpAct cga 17545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-eqg 17416 df-ga 17546 |
This theorem is referenced by: sylow1lem5 17840 sylow2alem2 17856 sylow3lem3 17867 |
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