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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 2738 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝐺)) | |
2 | eqidd 2738 | . 2 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝐺)) | |
3 | symggrp.1 | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
4 | eqid 2737 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | eqid 2737 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 3, 4, 5 | symgcl 19167 | . . 3 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
7 | 6 | 3adant1 1131 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
8 | 3, 4, 5 | symgcl 19167 | . . . 4 ⊢ ((𝑓 ∈ (Base‘𝐺) ∧ 𝑔 ∈ (Base‘𝐺)) → (𝑓(+g‘𝐺)𝑔) ∈ (Base‘𝐺)) |
9 | 3, 4, 5 | symgov 19166 | . . . 4 ⊢ ((𝑓 ∈ (Base‘𝐺) ∧ 𝑔 ∈ (Base‘𝐺)) → (𝑓(+g‘𝐺)𝑔) = (𝑓 ∘ 𝑔)) |
10 | 8, 9 | symggrplem 18695 | . . 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 483 | . 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 19168 | . 2 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺)) |
13 | 3, 4, 5 | symgov 19166 | . . . 4 ⊢ ((( I ↾ 𝐴) ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (( I ↾ 𝐴)(+g‘𝐺)𝑥) = (( I ↾ 𝐴) ∘ 𝑥)) |
14 | 12, 13 | sylan 581 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (( I ↾ 𝐴)(+g‘𝐺)𝑥) = (( I ↾ 𝐴) ∘ 𝑥)) |
15 | 3, 4 | elsymgbas 19156 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴–1-1-onto→𝐴)) |
16 | 15 | biimpa 478 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴–1-1-onto→𝐴) |
17 | f1of 6785 | . . . 4 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → 𝑥:𝐴⟶𝐴) | |
18 | fcoi2 6718 | . . . 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 2777 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (( I ↾ 𝐴)(+g‘𝐺)𝑥) = 𝑥) |
21 | f1ocnv 6797 | . . . . 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 19156 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (◡𝑥 ∈ (Base‘𝐺) ↔ ◡𝑥:𝐴–1-1-onto→𝐴)) |
24 | 22, 15, 23 | 3imtr4d 294 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) → ◡𝑥 ∈ (Base‘𝐺))) |
25 | 24 | imp 408 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ◡𝑥 ∈ (Base‘𝐺)) |
26 | 3, 4, 5 | symgov 19166 | . . . 4 ⊢ ((◡𝑥 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑥(+g‘𝐺)𝑥) = (◡𝑥 ∘ 𝑥)) |
27 | 25, 26 | sylancom 589 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑥(+g‘𝐺)𝑥) = (◡𝑥 ∘ 𝑥)) |
28 | f1ococnv1 6814 | . . . 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 2777 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑥(+g‘𝐺)𝑥) = ( I ↾ 𝐴)) |
31 | 1, 2, 7, 11, 12, 20, 25, 30 | isgrpd 18773 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 I cid 5531 ◡ccnv 5633 ↾ cres 5636 ∘ ccom 5638 ⟶wf 6493 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 +gcplusg 17134 Grpcgrp 18749 SymGrpcsymg 19149 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-tset 17153 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-efmnd 18680 df-grp 18752 df-symg 19150 |
This theorem is referenced by: symginv 19185 symgsubmefmndALT 19186 galactghm 19187 symgga 19190 pgrpsubgsymgbi 19191 pgrpsubgsymg 19192 idressubgsymg 19193 gsumccatsymgsn 19209 symgsssg 19250 symgfisg 19251 symggen 19253 symgtrinv 19255 psgnunilem5 19277 psgnunilem2 19278 psgnuni 19282 psgneldm2 19287 psgnfitr 19300 psgnghm 20987 zrhpsgninv 20992 evpmodpmf1o 21003 mdetleib2 21940 mdetdiag 21951 mdetralt 21960 mdetunilem7 21970 symgtgp 23460 symgfcoeu 31936 symgsubg 31941 cyc3co2 31992 cyc3genpmlem 32003 cyc3genpm 32004 cycpmconjs 32008 cyc3conja 32009 madjusmdetlem3 32413 madjusmdetlem4 32414 pgrple2abl 46448 |
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