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Mirrors > Home > MPE Home > Th. List > symggen2 | Structured version Visualization version GIF version |
Description: A finite permutation group is generated by the transpositions, see also Theorem 3.4 in [Rotman] p. 31. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
Ref | Expression |
---|---|
symgtrf.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
symgtrf.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
symgtrf.b | ⊢ 𝐵 = (Base‘𝐺) |
symggen.k | ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺)) |
Ref | Expression |
---|---|
symggen2 | ⊢ (𝐷 ∈ Fin → (𝐾‘𝑇) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgtrf.t | . . 3 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
2 | symgtrf.g | . . 3 ⊢ 𝐺 = (SymGrp‘𝐷) | |
3 | symgtrf.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
4 | symggen.k | . . 3 ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺)) | |
5 | 1, 2, 3, 4 | symggen 18247 | . 2 ⊢ (𝐷 ∈ Fin → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
6 | difss 3966 | . . . . . . 7 ⊢ (𝑥 ∖ I ) ⊆ 𝑥 | |
7 | dmss 5559 | . . . . . . 7 ⊢ ((𝑥 ∖ I ) ⊆ 𝑥 → dom (𝑥 ∖ I ) ⊆ dom 𝑥) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ dom (𝑥 ∖ I ) ⊆ dom 𝑥 |
9 | 2, 3 | symgbasf1o 18160 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → 𝑥:𝐷–1-1-onto→𝐷) |
10 | f1odm 6386 | . . . . . . 7 ⊢ (𝑥:𝐷–1-1-onto→𝐷 → dom 𝑥 = 𝐷) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → dom 𝑥 = 𝐷) |
12 | 8, 11 | syl5sseq 3878 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → dom (𝑥 ∖ I ) ⊆ 𝐷) |
13 | ssfi 8455 | . . . . 5 ⊢ ((𝐷 ∈ Fin ∧ dom (𝑥 ∖ I ) ⊆ 𝐷) → dom (𝑥 ∖ I ) ∈ Fin) | |
14 | 12, 13 | sylan2 586 | . . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ 𝐵) → dom (𝑥 ∖ I ) ∈ Fin) |
15 | 14 | ralrimiva 3175 | . . 3 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ 𝐵 dom (𝑥 ∖ I ) ∈ Fin) |
16 | rabid2 3329 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ↔ ∀𝑥 ∈ 𝐵 dom (𝑥 ∖ I ) ∈ Fin) | |
17 | 15, 16 | sylibr 226 | . 2 ⊢ (𝐷 ∈ Fin → 𝐵 = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
18 | 5, 17 | eqtr4d 2864 | 1 ⊢ (𝐷 ∈ Fin → (𝐾‘𝑇) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ∀wral 3117 {crab 3121 ∖ cdif 3795 ⊆ wss 3798 I cid 5251 dom cdm 5346 ran crn 5347 –1-1-onto→wf1o 6126 ‘cfv 6127 Fincfn 8228 Basecbs 16229 mrClscmrc 16603 SubMndcsubmnd 17694 SymGrpcsymg 18154 pmTrspcpmtr 18218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-tset 16331 df-0g 16462 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-subg 17949 df-symg 18155 df-pmtr 18219 |
This theorem is referenced by: psgnfitr 18295 mdetunilem7 20799 |
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