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Mirrors > Home > MPE Home > Th. List > psgnfitr | Structured version Visualization version GIF version |
Description: A permutation of a finite set is generated by transpositions. (Contributed by AV, 13-Jan-2019.) |
Ref | Expression |
---|---|
psgnfitr.g | ⊢ 𝐺 = (SymGrp‘𝑁) |
psgnfitr.p | ⊢ 𝐵 = (Base‘𝐺) |
psgnfitr.t | ⊢ 𝑇 = ran (pmTrsp‘𝑁) |
Ref | Expression |
---|---|
psgnfitr | ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝐵 ↔ ∃𝑤 ∈ Word 𝑇𝑄 = (𝐺 Σg 𝑤))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgnfitr.t | . . . . 5 ⊢ 𝑇 = ran (pmTrsp‘𝑁) | |
2 | psgnfitr.g | . . . . 5 ⊢ 𝐺 = (SymGrp‘𝑁) | |
3 | psgnfitr.p | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
4 | eqid 2825 | . . . . 5 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺)) | |
5 | 1, 2, 3, 4 | symggen2 18248 | . . . 4 ⊢ (𝑁 ∈ Fin → ((mrCls‘(SubMnd‘𝐺))‘𝑇) = 𝐵) |
6 | 2 | symggrp 18177 | . . . . . 6 ⊢ (𝑁 ∈ Fin → 𝐺 ∈ Grp) |
7 | grpmnd 17790 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ Fin → 𝐺 ∈ Mnd) |
9 | eqid 2825 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
10 | 1, 2, 9 | symgtrf 18246 | . . . . 5 ⊢ 𝑇 ⊆ (Base‘𝐺) |
11 | 9, 4 | gsumwspan 17744 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺)) → ((mrCls‘(SubMnd‘𝐺))‘𝑇) = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σg 𝑤))) |
12 | 8, 10, 11 | sylancl 580 | . . . 4 ⊢ (𝑁 ∈ Fin → ((mrCls‘(SubMnd‘𝐺))‘𝑇) = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σg 𝑤))) |
13 | 5, 12 | eqtr3d 2863 | . . 3 ⊢ (𝑁 ∈ Fin → 𝐵 = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σg 𝑤))) |
14 | 13 | eleq2d 2892 | . 2 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝐵 ↔ 𝑄 ∈ ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σg 𝑤)))) |
15 | eqid 2825 | . . 3 ⊢ (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σg 𝑤)) = (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σg 𝑤)) | |
16 | ovex 6942 | . . 3 ⊢ (𝐺 Σg 𝑤) ∈ V | |
17 | 15, 16 | elrnmpti 5613 | . 2 ⊢ (𝑄 ∈ ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σg 𝑤)) ↔ ∃𝑤 ∈ Word 𝑇𝑄 = (𝐺 Σg 𝑤)) |
18 | 14, 17 | syl6bb 279 | 1 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝐵 ↔ ∃𝑤 ∈ Word 𝑇𝑄 = (𝐺 Σg 𝑤))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 ⊆ wss 3798 ↦ cmpt 4954 ran crn 5347 ‘cfv 6127 (class class class)co 6910 Fincfn 8228 Word cword 13581 Basecbs 16229 Σg cgsu 16461 mrClscmrc 16603 Mndcmnd 17654 SubMndcsubmnd 17694 Grpcgrp 17783 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-fzo 12768 df-seq 13103 df-hash 13418 df-word 13582 df-concat 13638 df-s1 13663 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-gsum 16463 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: psgnfix1 20311 psgnfix2 20312 |
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