Theorem cayleyth 19384

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 2737	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2737	. . . 4 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎)))
5		eqid 2737	. . . 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 19383	. . 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 1143	. 2 ⊢ (𝐺 ∈ Grp → ran (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(+g‘𝐺)𝑎))) ∈ (SubGrp‘𝐻))
8	6	simp2d 1144	. . 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 1145	. . 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 6763	. . . 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 3565	. . 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 585	. 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 7369	. . . . 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 7377	. . . 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 6765	. . . 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 3313	. . 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 3565	. 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 585	1 ⊢ (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑠))𝑓:𝑋–1-1-onto→𝑠)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ↦ cmpt 5167  ran crn 5626  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361  Basecbs 17173   ↾_s cress 17194  +_gcplusg 17214  Grpcgrp 18903  SubGrpcsubg 19090   GrpHom cghm 19181  SymGrpcsymg 19338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-uz 12783  df-fz 13456  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-tset 17233  df-0g 17398  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-efmnd 18831  df-grp 18906  df-minusg 18907  df-sbg 18908  df-subg 19093  df-ghm 19182  df-ga 19259  df-symg 19339

This theorem is referenced by: (None)