Theorem cayleyth 19401

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 (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠)

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.

Step	Hyp	Ref	Expression
1		cayley.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
2		cayley.h	. . . 4 ⊢ 𝐻 = (SymGrp‘𝑋)
3		eqid 2734	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2734	. . . 4 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))
5		eqid 2734	. . . 4 ⊢ ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))
6	1, 2, 3, 4, 5	cayley 19400	. . 3 ⊢ (𝐺 ∈ Grp → (ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (SubGrp‘𝐻) ∧ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))) ∧ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))):𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))
7	6	simp1d 1142	. 2 ⊢ (𝐺 ∈ Grp → ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (SubGrp‘𝐻))
8	6	simp2d 1143	. . 3 ⊢ (𝐺 ∈ Grp → (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))))
9	6	simp3d 1144	. . 3 ⊢ (𝐺 ∈ Grp → (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))):𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))
10		f1oeq1 6816	. . . 4 ⊢ (𝑓 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ↔ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))):𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))
11	10	rspcev 3605	. . 3 ⊢ (((𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))) ∧ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))):𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))) → ∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))
12	8, 9, 11	syl2anc 584	. 2 ⊢ (𝐺 ∈ Grp → ∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))
13		oveq2 7421	. . . . 5 ⊢ (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (𝐻 ↾s 𝑠) = (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))
14	13	oveq2d 7429	. . . 4 ⊢ (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (𝐺 GrpHom (𝐻 ↾s 𝑠)) = (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))))))
15		f1oeq3 6818	. . . 4 ⊢ (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (𝑓:𝑋–1-1-onto→𝑠 ↔ 𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))
16	14, 15	rexeqbidv 3330	. . 3 ⊢ (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) → (∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠 ↔ ∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))
17	16	rspcev 3605	. 2 ⊢ ((ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (SubGrp‘𝐻) ∧ ∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))))𝑓:𝑋–1-1-onto→ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))) → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠)
18	7, 12, 17	syl2anc 584	1 ⊢ (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ∃wrex 3059   ↦ cmpt 5205  ran crn 5666  –1-1-onto→wf1o 6540  ‘cfv 6541  (class class class)co 7413  Basecbs 17229   ↾_s cress 17252  +_gcplusg 17273  Grpcgrp 18920  SubGrpcsubg 19107   GrpHom cghm 19199  SymGrpcsymg 19354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-map 8850  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-z 12597  df-uz 12861  df-fz 13530  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17230  df-ress 17253  df-plusg 17286  df-tset 17292  df-0g 17457  df-mgm 18622  df-sgrp 18701  df-mnd 18717  df-mhm 18765  df-submnd 18766  df-efmnd 18851  df-grp 18923  df-minusg 18924  df-sbg 18925  df-subg 19110  df-ghm 19200  df-ga 19277  df-symg 19355

This theorem is referenced by: (None)