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 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 2817	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2817	. . . 4 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))
5		eqid 2817	. . . 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 18055	. . 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 1165	. 2 ⊢ (𝐺 ∈ Grp → ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (SubGrp‘𝐻))
8	6	simp2d 1166	. . 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 1167	. . 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 6353	. . . 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 3513	. . 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 575	. 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 6892	. . . . 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 6900	. . . 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 6355	. . . 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 3353	. . 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 3513	. 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 575	1 ⊢ (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠)

Syntax hints:   → wi 4   = wceq 1637   ∈ wcel 2157  ∃wrex 3108   ↦ cmpt 4934  ran crn 5325  –1-1-onto→wf1o 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)