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Theorem cayleyth 18038
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 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.)
Hypotheses
Ref Expression
cayley.x 𝑋 = (Base‘𝐺)
cayley.h 𝐻 = (SymGrp‘𝑋)
Assertion
Ref Expression
cayleyth (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻s 𝑠))𝑓:𝑋1-1-onto𝑠)
Distinct variable groups:   𝑓,𝑠,𝐺   𝑓,𝐻,𝑠   𝑓,𝑋,𝑠

Proof of Theorem cayleyth
Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cayley.x . . . 4 𝑋 = (Base‘𝐺)
2 cayley.h . . . 4 𝐻 = (SymGrp‘𝑋)
3 eqid 2771 . . . 4 (+g𝐺) = (+g𝐺)
4 eqid 2771 . . . 4 (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
5 eqid 2771 . . . 4 ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
61, 2, 3, 4, 5cayley 18037 . . 3 (𝐺 ∈ Grp → (ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (SubGrp‘𝐻) ∧ (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))) ∧ (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))):𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
76simp1d 1136 . 2 (𝐺 ∈ Grp → ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (SubGrp‘𝐻))
86simp2d 1137 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
96simp3d 1138 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))):𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))
10 f1oeq1 6266 . . . 4 (𝑓 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ↔ (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))):𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1110rspcev 3460 . . 3 (((𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))) ∧ (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))):𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))) → ∃𝑓 ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))
128, 9, 11syl2anc 573 . 2 (𝐺 ∈ Grp → ∃𝑓 ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))
13 oveq2 6800 . . . . 5 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐻s 𝑠) = (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1413oveq2d 6808 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐺 GrpHom (𝐻s 𝑠)) = (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
15 f1oeq3 6268 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝑓:𝑋1-1-onto𝑠𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1614, 15rexeqbidv 3302 . . 3 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (∃𝑓 ∈ (𝐺 GrpHom (𝐻s 𝑠))𝑓:𝑋1-1-onto𝑠 ↔ ∃𝑓 ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1716rspcev 3460 . 2 ((ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (SubGrp‘𝐻) ∧ ∃𝑓 ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))) → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻s 𝑠))𝑓:𝑋1-1-onto𝑠)
187, 12, 17syl2anc 573 1 (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻s 𝑠))𝑓:𝑋1-1-onto𝑠)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wrex 3062  cmpt 4863  ran crn 5250  1-1-ontowf1o 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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