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Theorem cayleyth 18539
 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.)
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 2798 . . . 4 (+g𝐺) = (+g𝐺)
4 eqid 2798 . . . 4 (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
5 eqid 2798 . . . 4 ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
61, 2, 3, 4, 5cayley 18538 . . 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 1139 . 2 (𝐺 ∈ Grp → ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (SubGrp‘𝐻))
86simp2d 1140 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
96simp3d 1141 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))):𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))
10 f1oeq1 6580 . . . 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 3571 . . 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 587 . 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 7144 . . . . 5 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐻s 𝑠) = (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1413oveq2d 7152 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐺 GrpHom (𝐻s 𝑠)) = (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
15 f1oeq3 6582 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝑓:𝑋1-1-onto𝑠𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1614, 15rexeqbidv 3355 . . 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 3571 . 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 587 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 1538   ∈ wcel 2111  ∃wrex 3107   ↦ cmpt 5111  ran crn 5521  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136  Basecbs 16478   ↾s cress 16479  +gcplusg 16560  Grpcgrp 18098  SubGrpcsubg 18269   GrpHom cghm 18351  SymGrpcsymg 18491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-3 11692  df-4 11693  df-5 11694  df-6 11695  df-7 11696  df-8 11697  df-9 11698  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-tset 16579  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-efmnd 18029  df-grp 18101  df-minusg 18102  df-sbg 18103  df-subg 18272  df-ghm 18352  df-ga 18416  df-symg 18492 This theorem is referenced by: (None)
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