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Theorem cayleyth 19202
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 2733 . . . 4 (+gβ€˜πΊ) = (+gβ€˜πΊ)
4 eqid 2733 . . . 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 2733 . . . 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 19201 . . 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 1143 . 2 (𝐺 ∈ Grp β†’ ran (𝑔 ∈ 𝑋 ↦ (π‘Ž ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)π‘Ž))) ∈ (SubGrpβ€˜π»))
86simp2d 1144 . . 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 1145 . . 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 6773 . . . 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 3580 . . 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 585 . 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 7366 . . . . 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 7374 . . . 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 6775 . . . 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 3319 . . 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 3580 . 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 585 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 1542   ∈ wcel 2107  βˆƒwrex 3070   ↦ cmpt 5189  ran crn 5635  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088   β†Ύs cress 17117  +gcplusg 17138  Grpcgrp 18753  SubGrpcsubg 18927   GrpHom cghm 19010  SymGrpcsymg 19153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-tset 17157  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-submnd 18607  df-efmnd 18684  df-grp 18756  df-minusg 18757  df-sbg 18758  df-subg 18930  df-ghm 19011  df-ga 19075  df-symg 19154
This theorem is referenced by: (None)
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