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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 | β’ π = ran (pmTrspβπ·) |
symgtrf.g | β’ πΊ = (SymGrpβπ·) |
symgtrf.b | β’ π΅ = (BaseβπΊ) |
symggen.k | β’ πΎ = (mrClsβ(SubMndβπΊ)) |
Ref | Expression |
---|---|
symggen2 | β’ (π· β Fin β (πΎβπ) = π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgtrf.t | . . 3 β’ π = ran (pmTrspβπ·) | |
2 | symgtrf.g | . . 3 β’ πΊ = (SymGrpβπ·) | |
3 | symgtrf.b | . . 3 β’ π΅ = (BaseβπΊ) | |
4 | symggen.k | . . 3 β’ πΎ = (mrClsβ(SubMndβπΊ)) | |
5 | 1, 2, 3, 4 | symggen 19260 | . 2 β’ (π· β Fin β (πΎβπ) = {π₯ β π΅ β£ dom (π₯ β I ) β Fin}) |
6 | difss 4095 | . . . . . . 7 β’ (π₯ β I ) β π₯ | |
7 | dmss 5862 | . . . . . . 7 β’ ((π₯ β I ) β π₯ β dom (π₯ β I ) β dom π₯) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 β’ dom (π₯ β I ) β dom π₯ |
9 | 2, 3 | symgbasf1o 19164 | . . . . . . 7 β’ (π₯ β π΅ β π₯:π·β1-1-ontoβπ·) |
10 | f1odm 6792 | . . . . . . 7 β’ (π₯:π·β1-1-ontoβπ· β dom π₯ = π·) | |
11 | 9, 10 | syl 17 | . . . . . 6 β’ (π₯ β π΅ β dom π₯ = π·) |
12 | 8, 11 | sseqtrid 4000 | . . . . 5 β’ (π₯ β π΅ β dom (π₯ β I ) β π·) |
13 | ssfi 9123 | . . . . 5 β’ ((π· β Fin β§ dom (π₯ β I ) β π·) β dom (π₯ β I ) β Fin) | |
14 | 12, 13 | sylan2 594 | . . . 4 β’ ((π· β Fin β§ π₯ β π΅) β dom (π₯ β I ) β Fin) |
15 | 14 | ralrimiva 3140 | . . 3 β’ (π· β Fin β βπ₯ β π΅ dom (π₯ β I ) β Fin) |
16 | rabid2 3438 | . . 3 β’ (π΅ = {π₯ β π΅ β£ dom (π₯ β I ) β Fin} β βπ₯ β π΅ dom (π₯ β I ) β Fin) | |
17 | 15, 16 | sylibr 233 | . 2 β’ (π· β Fin β π΅ = {π₯ β π΅ β£ dom (π₯ β I ) β Fin}) |
18 | 5, 17 | eqtr4d 2776 | 1 β’ (π· β Fin β (πΎβπ) = π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwral 3061 {crab 3406 β cdif 3911 β wss 3914 I cid 5534 dom cdm 5637 ran crn 5638 β1-1-ontoβwf1o 6499 βcfv 6500 Fincfn 8889 Basecbs 17091 mrClscmrc 17471 SubMndcsubmnd 18608 SymGrpcsymg 19156 pmTrspcpmtr 19231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-tset 17160 df-0g 17331 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-efmnd 18687 df-grp 18759 df-minusg 18760 df-subg 18933 df-symg 19157 df-pmtr 19232 |
This theorem is referenced by: psgnfitr 19307 mdetunilem7 21990 |
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