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Theorem cayleyth 17607
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 2609 . . . 4 (+g𝐺) = (+g𝐺)
4 eqid 2609 . . . 4 (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
5 eqid 2609 . . . 4 ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
61, 2, 3, 4, 5cayley 17606 . . 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 1065 . 2 (𝐺 ∈ Grp → ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (SubGrp‘𝐻))
86simp2d 1066 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
96simp3d 1067 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))):𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))
10 f1oeq1 6025 . . . 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 3281 . . 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 690 . 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 6535 . . . . 5 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐻s 𝑠) = (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1413oveq2d 6543 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐺 GrpHom (𝐻s 𝑠)) = (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
15 f1oeq3 6027 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝑓:𝑋1-1-onto𝑠𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1614, 15rexeqbidv 3129 . . 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 3281 . 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 690 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 1474  wcel 1976  wrex 2896  cmpt 4637  ran crn 5029  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  Basecbs 15644  s cress 15645  +gcplusg 15717  Grpcgrp 17194  SubGrpcsubg 17360   GrpHom cghm 17429  SymGrpcsymg 17569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-plusg 15730  df-tset 15736  df-0g 15874  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-mhm 17107  df-submnd 17108  df-grp 17197  df-minusg 17198  df-sbg 17199  df-subg 17363  df-ghm 17430  df-ga 17495  df-symg 17570
This theorem is referenced by: (None)
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