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Theorem cayley 14805
Description: Cayley's Theorem (constructive version): given group 
G,  F is an isomorphism between  G and the subgroup  S of the symmetry group  H on the underlying set  X of  G. (Contributed by Paul Chapman, 3-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
cayley.x  |-  X  =  ( Base `  G
)
cayley.h  |-  H  =  ( SymGrp `  X )
cayley.p  |-  .+  =  ( +g  `  G )
cayley.f  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
cayley.s  |-  S  =  ran  F
Assertion
Ref Expression
cayley  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  H
)  /\  F  e.  ( G  GrpHom  ( Hs  S ) )  /\  F : X -1-1-onto-> S ) )
Distinct variable groups:    g, a, G    g, H    .+ , a, g    X, a, g
Allowed substitution hints:    S( g, a)    F( g, a)    H( a)

Proof of Theorem cayley
StepHypRef Expression
1 cayley.s . . 3  |-  S  =  ran  F
2 cayley.x . . . . 5  |-  X  =  ( Base `  G
)
3 cayley.p . . . . 5  |-  .+  =  ( +g  `  G )
4 eqid 2296 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
5 cayley.h . . . . 5  |-  H  =  ( SymGrp `  X )
6 eqid 2296 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
7 cayley.f . . . . 5  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
82, 3, 4, 5, 6, 7cayleylem1 14803 . . . 4  |-  ( G  e.  Grp  ->  F  e.  ( G  GrpHom  H ) )
9 ghmrn 14712 . . . 4  |-  ( F  e.  ( G  GrpHom  H )  ->  ran  F  e.  (SubGrp `  H )
)
108, 9syl 15 . . 3  |-  ( G  e.  Grp  ->  ran  F  e.  (SubGrp `  H
) )
111, 10syl5eqel 2380 . 2  |-  ( G  e.  Grp  ->  S  e.  (SubGrp `  H )
)
121eqimss2i 3246 . . . 4  |-  ran  F  C_  S
13 eqid 2296 . . . . 5  |-  ( Hs  S )  =  ( Hs  S )
1413resghm2b 14717 . . . 4  |-  ( ( S  e.  (SubGrp `  H )  /\  ran  F 
C_  S )  -> 
( F  e.  ( G  GrpHom  H )  <->  F  e.  ( G  GrpHom  ( Hs  S ) ) ) )
1511, 12, 14sylancl 643 . . 3  |-  ( G  e.  Grp  ->  ( F  e.  ( G  GrpHom  H )  <->  F  e.  ( G  GrpHom  ( Hs  S ) ) ) )
168, 15mpbid 201 . 2  |-  ( G  e.  Grp  ->  F  e.  ( G  GrpHom  ( Hs  S ) ) )
172, 3, 4, 5, 6, 7cayleylem2 14804 . . . 4  |-  ( G  e.  Grp  ->  F : X -1-1-> ( Base `  H
) )
18 f1f1orn 5499 . . . 4  |-  ( F : X -1-1-> ( Base `  H )  ->  F : X -1-1-onto-> ran  F )
1917, 18syl 15 . . 3  |-  ( G  e.  Grp  ->  F : X -1-1-onto-> ran  F )
20 f1oeq3 5481 . . . 4  |-  ( S  =  ran  F  -> 
( F : X -1-1-onto-> S  <->  F : X -1-1-onto-> ran  F ) )
211, 20ax-mp 8 . . 3  |-  ( F : X -1-1-onto-> S  <->  F : X -1-1-onto-> ran  F
)
2219, 21sylibr 203 . 2  |-  ( G  e.  Grp  ->  F : X -1-1-onto-> S )
2311, 16, 223jca 1132 1  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  H
)  /\  F  e.  ( G  GrpHom  ( Hs  S ) )  /\  F : X -1-1-onto-> S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165    e. cmpt 4093   ran crn 4706   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378  SubGrpcsubg 14631    GrpHom cghm 14696   SymGrpcsymg 14785
This theorem is referenced by:  cayleyth  14806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-tset 13243  df-0g 13420  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-ga 14760  df-symg 14786
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