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Theorem cayleyth 18056
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 2817 . . . 4 (+g𝐺) = (+g𝐺)
4 eqid 2817 . . . 4 (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
5 eqid 2817 . . . 4 ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
61, 2, 3, 4, 5cayley 18055 . . 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 1165 . 2 (𝐺 ∈ Grp → ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (SubGrp‘𝐻))
86simp2d 1166 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
96simp3d 1167 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))):𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))
10 f1oeq1 6353 . . . 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 3513 . . 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 575 . 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 6892 . . . . 5 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐻s 𝑠) = (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1413oveq2d 6900 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐺 GrpHom (𝐻s 𝑠)) = (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
15 f1oeq3 6355 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝑓:𝑋1-1-onto𝑠𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1614, 15rexeqbidv 3353 . . 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 3513 . 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 575 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 1637  wcel 2157  wrex 3108  cmpt 4934  ran crn 5325  1-1-ontowf1o 6110  cfv 6111  (class class class)co 6884  Basecbs 16088  s cress 16089  +gcplusg 16173  Grpcgrp 17647  SubGrpcsubg 17810   GrpHom cghm 17879  SymGrpcsymg 18018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189  ax-cnex 10287  ax-resscn 10288  ax-1cn 10289  ax-icn 10290  ax-addcl 10291  ax-addrcl 10292  ax-mulcl 10293  ax-mulrcl 10294  ax-mulcom 10295  ax-addass 10296  ax-mulass 10297  ax-distr 10298  ax-i2m1 10299  ax-1ne0 10300  ax-1rid 10301  ax-rnegex 10302  ax-rrecex 10303  ax-cnre 10304  ax-pre-lttri 10305  ax-pre-lttrn 10306  ax-pre-ltadd 10307  ax-pre-mulgt0 10308
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-om 7306  df-1st 7408  df-2nd 7409  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-1o 7806  df-oadd 7810  df-er 7989  df-map 8104  df-en 8203  df-dom 8204  df-sdom 8205  df-fin 8206  df-pnf 10371  df-mnf 10372  df-xr 10373  df-ltxr 10374  df-le 10375  df-sub 10563  df-neg 10564  df-nn 11316  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-n0 11580  df-z 11664  df-uz 11925  df-fz 12570  df-struct 16090  df-ndx 16091  df-slot 16092  df-base 16094  df-sets 16095  df-ress 16096  df-plusg 16186  df-tset 16192  df-0g 16327  df-mgm 17467  df-sgrp 17509  df-mnd 17520  df-mhm 17560  df-submnd 17561  df-grp 17650  df-minusg 17651  df-sbg 17652  df-subg 17813  df-ghm 17880  df-ga 17944  df-symg 18019
This theorem is referenced by: (None)
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