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Mirrors > Home > MPE Home > Th. List > cayleyth | Structured version Visualization version GIF version |
Description: Cayley's Theorem (existence version): every group 𝐺 is isomorphic to a subgroup of the symmetric group on the underlying set of 𝐺. (For any group 𝐺 there exists an isomorphism 𝑓 between 𝐺 and a subgroup ℎ of the symmetric group on the underlying set of 𝐺.) See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
cayley.x | ⊢ 𝑋 = (Base‘𝐺) |
cayley.h | ⊢ 𝐻 = (SymGrp‘𝑋) |
Ref | Expression |
---|---|
cayleyth | ⊢ (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cayley.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
2 | cayley.h | . . . 4 ⊢ 𝐻 = (SymGrp‘𝑋) | |
3 | eqid 2771 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2771 | . . . 4 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) | |
5 | eqid 2771 | . . . 4 ⊢ ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) | |
6 | 1, 2, 3, 4, 5 | cayley 18037 | . . 3 ⊢ (𝐺 ∈ Grp → (ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (SubGrp‘𝐻) ∧ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))) ∧ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))):𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))) |
7 | 6 | simp1d 1136 | . 2 ⊢ (𝐺 ∈ Grp → ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (SubGrp‘𝐻)) |
8 | 6 | simp2d 1137 | . . 3 ⊢ (𝐺 ∈ Grp → (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))) |
9 | 6 | simp3d 1138 | . . 3 ⊢ (𝐺 ∈ Grp → (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))):𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))) |
10 | f1oeq1 6266 | . . . 4 ⊢ (𝑓 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ↔ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))):𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))) | |
11 | 10 | rspcev 3460 | . . 3 ⊢ (((𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))) ∧ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))):𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))) → ∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))) |
12 | 8, 9, 11 | syl2anc 573 | . 2 ⊢ (𝐺 ∈ Grp → ∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))) |
13 | oveq2 6800 | . . . . 5 ⊢ (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (𝐻 ↾s 𝑠) = (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))) | |
14 | 13 | oveq2d 6808 | . . . 4 ⊢ (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (𝐺 GrpHom (𝐻 ↾s 𝑠)) = (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))) |
15 | f1oeq3 6268 | . . . 4 ⊢ (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (𝑓:𝑋–1-1-onto→𝑠 ↔ 𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))) | |
16 | 14, 15 | rexeqbidv 3302 | . . 3 ⊢ (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠 ↔ ∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))) |
17 | 16 | rspcev 3460 | . 2 ⊢ ((ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (SubGrp‘𝐻) ∧ ∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))) → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠) |
18 | 7, 12, 17 | syl2anc 573 | 1 ⊢ (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ∃wrex 3062 ↦ cmpt 4863 ran crn 5250 –1-1-onto→wf1o 6028 ‘cfv 6029 (class class class)co 6792 Basecbs 16060 ↾s cress 16061 +gcplusg 16145 Grpcgrp 17626 SubGrpcsubg 17792 GrpHom cghm 17861 SymGrpcsymg 18000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11496 df-z 11581 df-uz 11890 df-fz 12530 df-struct 16062 df-ndx 16063 df-slot 16064 df-base 16066 df-sets 16067 df-ress 16068 df-plusg 16158 df-tset 16164 df-0g 16306 df-mgm 17446 df-sgrp 17488 df-mnd 17499 df-mhm 17539 df-submnd 17540 df-grp 17629 df-minusg 17630 df-sbg 17631 df-subg 17795 df-ghm 17862 df-ga 17926 df-symg 18001 |
This theorem is referenced by: (None) |
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