![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 19468 | . 2 ⊢ (𝐷 ∈ Fin → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
6 | difss 4131 | . . . . . . 7 ⊢ (𝑥 ∖ I ) ⊆ 𝑥 | |
7 | dmss 5909 | . . . . . . 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 19372 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → 𝑥:𝐷–1-1-onto→𝐷) |
10 | f1odm 6847 | . . . . . . 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 4032 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → dom (𝑥 ∖ I ) ⊆ 𝐷) |
13 | ssfi 9211 | . . . . 5 ⊢ ((𝐷 ∈ Fin ∧ dom (𝑥 ∖ I ) ⊆ 𝐷) → dom (𝑥 ∖ I ) ∈ Fin) | |
14 | 12, 13 | sylan2 591 | . . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ 𝐵) → dom (𝑥 ∖ I ) ∈ Fin) |
15 | 14 | ralrimiva 3136 | . . 3 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ 𝐵 dom (𝑥 ∖ I ) ∈ Fin) |
16 | rabid2 3453 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ↔ ∀𝑥 ∈ 𝐵 dom (𝑥 ∖ I ) ∈ Fin) | |
17 | 15, 16 | sylibr 233 | . 2 ⊢ (𝐷 ∈ Fin → 𝐵 = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
18 | 5, 17 | eqtr4d 2769 | 1 ⊢ (𝐷 ∈ Fin → (𝐾‘𝑇) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3051 {crab 3419 ∖ cdif 3944 ⊆ wss 3947 I cid 5579 dom cdm 5682 ran crn 5683 –1-1-onto→wf1o 6553 ‘cfv 6554 Fincfn 8974 Basecbs 17213 mrClscmrc 17596 SubMndcsubmnd 18772 SymGrpcsymg 19364 pmTrspcpmtr 19439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-tset 17285 df-0g 17456 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-efmnd 18859 df-grp 18931 df-minusg 18932 df-subg 19117 df-symg 19365 df-pmtr 19440 |
This theorem is referenced by: psgnfitr 19515 mdetunilem7 22611 |
Copyright terms: Public domain | W3C validator |