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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 18590 | . 2 ⊢ (𝐷 ∈ Fin → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
6 | difss 4059 | . . . . . . 7 ⊢ (𝑥 ∖ I ) ⊆ 𝑥 | |
7 | dmss 5735 | . . . . . . 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 18495 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → 𝑥:𝐷–1-1-onto→𝐷) |
10 | f1odm 6594 | . . . . . . 7 ⊢ (𝑥:𝐷–1-1-onto→𝐷 → dom 𝑥 = 𝐷) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → dom 𝑥 = 𝐷) |
12 | 8, 11 | sseqtrid 3967 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → dom (𝑥 ∖ I ) ⊆ 𝐷) |
13 | ssfi 8722 | . . . . 5 ⊢ ((𝐷 ∈ Fin ∧ dom (𝑥 ∖ I ) ⊆ 𝐷) → dom (𝑥 ∖ I ) ∈ Fin) | |
14 | 12, 13 | sylan2 595 | . . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ 𝐵) → dom (𝑥 ∖ I ) ∈ Fin) |
15 | 14 | ralrimiva 3149 | . . 3 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ 𝐵 dom (𝑥 ∖ I ) ∈ Fin) |
16 | rabid2 3334 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ↔ ∀𝑥 ∈ 𝐵 dom (𝑥 ∖ I ) ∈ Fin) | |
17 | 15, 16 | sylibr 237 | . 2 ⊢ (𝐷 ∈ Fin → 𝐵 = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
18 | 5, 17 | eqtr4d 2836 | 1 ⊢ (𝐷 ∈ Fin → (𝐾‘𝑇) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ∖ cdif 3878 ⊆ wss 3881 I cid 5424 dom cdm 5519 ran crn 5520 –1-1-onto→wf1o 6323 ‘cfv 6324 Fincfn 8492 Basecbs 16475 mrClscmrc 16846 SubMndcsubmnd 17947 SymGrpcsymg 18487 pmTrspcpmtr 18561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-tset 16576 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-efmnd 18026 df-grp 18098 df-minusg 18099 df-subg 18268 df-symg 18488 df-pmtr 18562 |
This theorem is referenced by: psgnfitr 18637 mdetunilem7 21223 |
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