MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cayleyth Structured version   Visualization version   GIF version

Theorem cayleyth 19335
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 β„Ž 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 = (Baseβ€˜πΊ)
cayley.h 𝐻 = (SymGrpβ€˜π‘‹)
Assertion
Ref Expression
cayleyth (𝐺 ∈ Grp β†’ βˆƒπ‘  ∈ (SubGrpβ€˜π»)βˆƒπ‘“ ∈ (𝐺 GrpHom (𝐻 β†Ύs 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 π‘Ž 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cayley.x . . . 4 𝑋 = (Baseβ€˜πΊ)
2 cayley.h . . . 4 𝐻 = (SymGrpβ€˜π‘‹)
3 eqid 2726 . . . 4 (+gβ€˜πΊ) = (+gβ€˜πΊ)
4 eqid 2726 . . . 4 (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) = (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))
5 eqid 2726 . . . 4 ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) = ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))
61, 2, 3, 4, 5cayley 19334 . . 3 (𝐺 ∈ Grp β†’ (ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) ∈ (SubGrpβ€˜π») ∧ (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) ∈ (𝐺 GrpHom (𝐻 β†Ύs ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))))) ∧ (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))):𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))))
76simp1d 1139 . 2 (𝐺 ∈ Grp β†’ ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) ∈ (SubGrpβ€˜π»))
86simp2d 1140 . . 3 (𝐺 ∈ Grp β†’ (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) ∈ (𝐺 GrpHom (𝐻 β†Ύs ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))))))
96simp3d 1141 . . 3 (𝐺 ∈ Grp β†’ (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))):𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))))
10 f1oeq1 6815 . . . 4 (𝑓 = (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) β†’ (𝑓:𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) ↔ (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))):𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))))
1110rspcev 3606 . . 3 (((𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) ∈ (𝐺 GrpHom (𝐻 β†Ύs ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))))) ∧ (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))):𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))) β†’ βˆƒπ‘“ ∈ (𝐺 GrpHom (𝐻 β†Ύs ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))))𝑓:𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))))
128, 9, 11syl2anc 583 . 2 (𝐺 ∈ Grp β†’ βˆƒπ‘“ ∈ (𝐺 GrpHom (𝐻 β†Ύs ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))))𝑓:𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))))
13 oveq2 7413 . . . . 5 (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) β†’ (𝐻 β†Ύs 𝑠) = (𝐻 β†Ύs ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))))
1413oveq2d 7421 . . . 4 (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) β†’ (𝐺 GrpHom (𝐻 β†Ύs 𝑠)) = (𝐺 GrpHom (𝐻 β†Ύs ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))))))
15 f1oeq3 6817 . . . 4 (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) β†’ (𝑓:𝑋–1-1-onto→𝑠 ↔ 𝑓:𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))))
1614, 15rexeqbidv 3337 . . 3 (𝑠 = ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) β†’ (βˆƒπ‘“ ∈ (𝐺 GrpHom (𝐻 β†Ύs 𝑠))𝑓:𝑋–1-1-onto→𝑠 ↔ βˆƒπ‘“ ∈ (𝐺 GrpHom (𝐻 β†Ύs ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))))𝑓:𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))))
1716rspcev 3606 . 2 ((ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) ∈ (SubGrpβ€˜π») ∧ βˆƒπ‘“ ∈ (𝐺 GrpHom (𝐻 β†Ύs ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))))𝑓:𝑋–1-1-ontoβ†’ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž)))) β†’ βˆƒπ‘  ∈ (SubGrpβ€˜π»)βˆƒπ‘“ ∈ (𝐺 GrpHom (𝐻 β†Ύs 𝑠))𝑓:𝑋–1-1-onto→𝑠)
187, 12, 17syl2anc 583 1 (𝐺 ∈ Grp β†’ βˆƒπ‘  ∈ (SubGrpβ€˜π»)βˆƒπ‘“ ∈ (𝐺 GrpHom (𝐻 β†Ύs 𝑠))𝑓:𝑋–1-1-onto→𝑠)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   ↦ cmpt 5224  ran crn 5670  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153   β†Ύs cress 17182  +gcplusg 17206  Grpcgrp 18863  SubGrpcsubg 19047   GrpHom cghm 19138  SymGrpcsymg 19286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-tset 17225  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-submnd 18714  df-efmnd 18794  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19050  df-ghm 19139  df-ga 19206  df-symg 19287
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator