Theorem cayleyth 19370

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 2728	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2728	. . . 4 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))
5		eqid 2728	. . . 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 19369	. . 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 1140	. 2 ⊢ (𝐺 ∈ Grp → ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (SubGrp‘𝐻))
8	6	simp2d 1141	. . 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 1142	. . 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 6827	. . . 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 3609	. . 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 583	. 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 7428	. . . . 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 7436	. . . 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 6829	. . . 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 3340	. . 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 3609	. 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 583	1 ⊢ (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ∃wrex 3067   ↦ cmpt 5231  ran crn 5679  –1-1-onto→wf1o 6547  ‘cfv 6548  (class class class)co 7420  Basecbs 17180   ↾_s cress 17209  +_gcplusg 17233  Grpcgrp 18890  SubGrpcsubg 19075   GrpHom cghm 19167  SymGrpcsymg 19321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-uz 12854  df-fz 13518  df-struct 17116  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-tset 17252  df-0g 17423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-mhm 18740  df-submnd 18741  df-efmnd 18821  df-grp 18893  df-minusg 18894  df-sbg 18895  df-subg 19078  df-ghm 19168  df-ga 19241  df-symg 19322

This theorem is referenced by: (None)

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