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Mirrors > Home > MPE Home > Th. List > symgtrf | Structured version Visualization version GIF version |
Description: Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
Ref | Expression |
---|---|
symgtrf.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
symgtrf.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
symgtrf.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
symgtrf | ⊢ 𝑇 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷) | |
2 | symgtrf.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
3 | 1, 2 | pmtrff1o 19323 | . . 3 ⊢ (𝑥 ∈ 𝑇 → 𝑥:𝐷–1-1-onto→𝐷) |
4 | symgtrf.g | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐷) | |
5 | symgtrf.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
6 | 4, 5 | elsymgbas2 19232 | . . 3 ⊢ (𝑥 ∈ 𝑇 → (𝑥 ∈ 𝐵 ↔ 𝑥:𝐷–1-1-onto→𝐷)) |
7 | 3, 6 | mpbird 257 | . 2 ⊢ (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝐵) |
8 | 7 | ssriv 3984 | 1 ⊢ 𝑇 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ⊆ wss 3946 ran crn 5675 –1-1-onto→wf1o 6538 ‘cfv 6539 Basecbs 17139 SymGrpcsymg 19226 pmTrspcpmtr 19301 |
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 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-2o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-tset 17211 df-efmnd 18745 df-symg 19227 df-pmtr 19302 |
This theorem is referenced by: symggen 19330 symgtrinv 19332 pmtrdifellem4 19339 psgnunilem5 19354 psgnunilem2 19355 psgnuni 19359 psgneldm2 19364 psgnpmtr 19370 psgnfitr 19377 psgnghm 21116 pmtrodpm 21133 psgndiflemB 21136 mdetralt 22091 mdetunilem7 22101 fzto1st 32239 psgnfzto1st 32241 cyc3genpmlem 32287 cyc3genpm 32288 pgrpgt2nabl 46943 |
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