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Mirrors > Home > MPE Home > Th. List > symggrp | Structured version Visualization version GIF version |
Description: The symmetric group on a set 𝐴 is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 28-Jan-2024.) |
Ref | Expression |
---|---|
symggrp.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
Ref | Expression |
---|---|
symggrp | ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2727 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝐺)) | |
2 | eqidd 2727 | . 2 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝐺)) | |
3 | symggrp.1 | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
4 | eqid 2726 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | eqid 2726 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 3, 4, 5 | symgcl 19378 | . . 3 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
7 | 6 | 3adant1 1127 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
8 | 3, 4, 5 | symgcl 19378 | . . . 4 ⊢ ((𝑓 ∈ (Base‘𝐺) ∧ 𝑔 ∈ (Base‘𝐺)) → (𝑓(+g‘𝐺)𝑔) ∈ (Base‘𝐺)) |
9 | 3, 4, 5 | symgov 19377 | . . . 4 ⊢ ((𝑓 ∈ (Base‘𝐺) ∧ 𝑔 ∈ (Base‘𝐺)) → (𝑓(+g‘𝐺)𝑔) = (𝑓 ∘ 𝑔)) |
10 | 8, 9 | symggrplem 18869 | . . 3 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))) |
11 | 10 | adantl 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))) |
12 | 3 | idresperm 19379 | . 2 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺)) |
13 | 3, 4, 5 | symgov 19377 | . . . 4 ⊢ ((( I ↾ 𝐴) ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (( I ↾ 𝐴)(+g‘𝐺)𝑥) = (( I ↾ 𝐴) ∘ 𝑥)) |
14 | 12, 13 | sylan 578 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (( I ↾ 𝐴)(+g‘𝐺)𝑥) = (( I ↾ 𝐴) ∘ 𝑥)) |
15 | 3, 4 | elsymgbas 19367 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴–1-1-onto→𝐴)) |
16 | 15 | biimpa 475 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴–1-1-onto→𝐴) |
17 | f1of 6835 | . . . 4 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → 𝑥:𝐴⟶𝐴) | |
18 | fcoi2 6769 | . . . 4 ⊢ (𝑥:𝐴⟶𝐴 → (( I ↾ 𝐴) ∘ 𝑥) = 𝑥) | |
19 | 16, 17, 18 | 3syl 18 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (( I ↾ 𝐴) ∘ 𝑥) = 𝑥) |
20 | 14, 19 | eqtrd 2766 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (( I ↾ 𝐴)(+g‘𝐺)𝑥) = 𝑥) |
21 | f1ocnv 6847 | . . . . 5 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → ◡𝑥:𝐴–1-1-onto→𝐴) | |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐴–1-1-onto→𝐴 → ◡𝑥:𝐴–1-1-onto→𝐴)) |
23 | 3, 4 | elsymgbas 19367 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (◡𝑥 ∈ (Base‘𝐺) ↔ ◡𝑥:𝐴–1-1-onto→𝐴)) |
24 | 22, 15, 23 | 3imtr4d 293 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) → ◡𝑥 ∈ (Base‘𝐺))) |
25 | 24 | imp 405 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ◡𝑥 ∈ (Base‘𝐺)) |
26 | 3, 4, 5 | symgov 19377 | . . . 4 ⊢ ((◡𝑥 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑥(+g‘𝐺)𝑥) = (◡𝑥 ∘ 𝑥)) |
27 | 25, 26 | sylancom 586 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑥(+g‘𝐺)𝑥) = (◡𝑥 ∘ 𝑥)) |
28 | f1ococnv1 6864 | . . . 4 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → (◡𝑥 ∘ 𝑥) = ( I ↾ 𝐴)) | |
29 | 16, 28 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑥 ∘ 𝑥) = ( I ↾ 𝐴)) |
30 | 27, 29 | eqtrd 2766 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑥(+g‘𝐺)𝑥) = ( I ↾ 𝐴)) |
31 | 1, 2, 7, 11, 12, 20, 25, 30 | isgrpd 18948 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 I cid 5571 ◡ccnv 5673 ↾ cres 5676 ∘ ccom 5678 ⟶wf 6542 –1-1-onto→wf1o 6545 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 +gcplusg 17261 Grpcgrp 18923 SymGrpcsymg 19360 |
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 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 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 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-tset 17280 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-efmnd 18854 df-grp 18926 df-symg 19361 |
This theorem is referenced by: symginv 19396 symgsubmefmndALT 19397 galactghm 19398 symgga 19401 pgrpsubgsymgbi 19402 pgrpsubgsymg 19403 idressubgsymg 19404 gsumccatsymgsn 19420 symgsssg 19461 symgfisg 19462 symggen 19464 symgtrinv 19466 psgnunilem5 19488 psgnunilem2 19489 psgnuni 19493 psgneldm2 19498 psgnfitr 19511 psgnghm 21572 zrhpsgninv 21577 evpmodpmf1o 21588 mdetleib2 22578 mdetdiag 22589 mdetralt 22598 mdetunilem7 22608 symgtgp 24098 symgfcoeu 32964 symgsubg 32969 cyc3co2 33022 cyc3genpmlem 33033 cyc3genpm 33034 cycpmconjs 33038 cyc3conja 33039 madjusmdetlem3 33657 madjusmdetlem4 33658 pgrple2abl 47780 |
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