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Theorem cayleyth 18304
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 2778 . . . 4 (+g𝐺) = (+g𝐺)
4 eqid 2778 . . . 4 (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
5 eqid 2778 . . . 4 ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))
61, 2, 3, 4, 5cayley 18303 . . 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 1122 . 2 (𝐺 ∈ Grp → ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (SubGrp‘𝐻))
86simp2d 1123 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
96simp3d 1124 . . 3 (𝐺 ∈ Grp → (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))):𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))
10 f1oeq1 6433 . . . 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 3535 . . 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 576 . 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 6984 . . . . 5 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐻s 𝑠) = (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1413oveq2d 6992 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝐺 GrpHom (𝐻s 𝑠)) = (𝐺 GrpHom (𝐻s ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))))))
15 f1oeq3 6435 . . . 4 (𝑠 = ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎))) → (𝑓:𝑋1-1-onto𝑠𝑓:𝑋1-1-onto→ran (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔(+g𝐺)𝑎)))))
1614, 15rexeqbidv 3342 . . 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 3535 . 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 576 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 1507  wcel 2050  wrex 3089  cmpt 5008  ran crn 5408  1-1-ontowf1o 6187  cfv 6188  (class class class)co 6976  Basecbs 16339  s cress 16340  +gcplusg 16421  Grpcgrp 17891  SubGrpcsubg 18057   GrpHom cghm 18126  SymGrpcsymg 18266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-2 11503  df-3 11504  df-4 11505  df-5 11506  df-6 11507  df-7 11508  df-8 11509  df-9 11510  df-n0 11708  df-z 11794  df-uz 12059  df-fz 12709  df-struct 16341  df-ndx 16342  df-slot 16343  df-base 16345  df-sets 16346  df-ress 16347  df-plusg 16434  df-tset 16440  df-0g 16571  df-mgm 17710  df-sgrp 17752  df-mnd 17763  df-mhm 17803  df-submnd 17804  df-grp 17894  df-minusg 17895  df-sbg 17896  df-subg 18060  df-ghm 18127  df-ga 18191  df-symg 18267
This theorem is referenced by: (None)
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