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Theorem cayleyth 16567
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  e.  Grp  ->  E. s  e.  (SubGrp `  H ) E. f  e.  ( G  GrpHom  ( Hs  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.
StepHypRef Expression
1 cayley.x . . . 4  |-  X  =  ( Base `  G
)
2 cayley.h . . . 4  |-  H  =  ( SymGrp `  X )
3 eqid 2457 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2457 . . . 4  |-  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )
5 eqid 2457 . . . 4  |-  ran  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )  =  ran  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )
61, 2, 3, 4, 5cayley 16566 . . 3  |-  ( G  e.  Grp  ->  ( ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) )  e.  (SubGrp `  H )  /\  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )  e.  ( G 
GrpHom  ( Hs  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) ) )  /\  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) ) : X -1-1-onto-> ran  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) ) ) )
76simp1d 1008 . 2  |-  ( G  e.  Grp  ->  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) )  e.  (SubGrp `  H ) )
86simp2d 1009 . . 3  |-  ( G  e.  Grp  ->  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )  e.  ( G 
GrpHom  ( Hs  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) ) ) )
96simp3d 1010 . . 3  |-  ( G  e.  Grp  ->  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) ) : X -1-1-onto-> ran  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) ) )
10 f1oeq1 5813 . . . 4  |-  ( f  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) )  ->  ( f : X -1-1-onto-> ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) )  <-> 
( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) : X -1-1-onto-> ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) ) )
1110rspcev 3210 . . 3  |-  ( ( ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) )  e.  ( G  GrpHom  ( Hs  ran  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) ) ) )  /\  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) : X -1-1-onto-> ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) )  ->  E. f  e.  ( G  GrpHom  ( Hs  ran  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) ) ) ) f : X -1-1-onto-> ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) )
128, 9, 11syl2anc 661 . 2  |-  ( G  e.  Grp  ->  E. f  e.  ( G  GrpHom  ( Hs  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) ) ) f : X -1-1-onto-> ran  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) ) )
13 oveq2 6304 . . . . 5  |-  ( s  =  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )  ->  ( Hs  s
)  =  ( Hs  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) ) )
1413oveq2d 6312 . . . 4  |-  ( s  =  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )  ->  ( G  GrpHom  ( Hs  s ) )  =  ( G  GrpHom  ( Hs 
ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) ) ) )
15 f1oeq3 5815 . . . 4  |-  ( s  =  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )  ->  ( f : X -1-1-onto-> s  <->  f : X -1-1-onto-> ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) ) )
1614, 15rexeqbidv 3069 . . 3  |-  ( s  =  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) )  ->  ( E. f  e.  ( G  GrpHom  ( Hs  s ) ) f : X -1-1-onto-> s  <->  E. f  e.  ( G  GrpHom  ( Hs  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) ) ) f : X -1-1-onto-> ran  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) ) ) )
1716rspcev 3210 . 2  |-  ( ( ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) )  e.  (SubGrp `  H
)  /\  E. f  e.  ( G  GrpHom  ( Hs  ran  ( g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G ) a ) ) ) ) ) f : X -1-1-onto-> ran  (
g  e.  X  |->  ( a  e.  X  |->  ( g ( +g  `  G
) a ) ) ) )  ->  E. s  e.  (SubGrp `  H ) E. f  e.  ( G  GrpHom  ( Hs  s ) ) f : X -1-1-onto-> s
)
187, 12, 17syl2anc 661 1  |-  ( G  e.  Grp  ->  E. s  e.  (SubGrp `  H ) E. f  e.  ( G  GrpHom  ( Hs  s ) ) f : X -1-1-onto-> s
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   E.wrex 2808    |-> cmpt 4515   ran crn 5009   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   Basecbs 14644   ↾s cress 14645   +g cplusg 14712   Grpcgrp 16180  SubGrpcsubg 16322    GrpHom cghm 16391   SymGrpcsymg 16529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-tset 14731  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-subg 16325  df-ghm 16392  df-ga 16455  df-symg 16530
This theorem is referenced by: (None)
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