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Theorem cayleylem2 14837
Description: Lemma for cayley 14838. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
cayleylem1.x  |-  X  =  ( Base `  G
)
cayleylem1.p  |-  .+  =  ( +g  `  G )
cayleylem1.u  |-  .0.  =  ( 0g `  G )
cayleylem1.h  |-  H  =  ( SymGrp `  X )
cayleylem1.s  |-  S  =  ( Base `  H
)
cayleylem1.f  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
Assertion
Ref Expression
cayleylem2  |-  ( G  e.  Grp  ->  F : X -1-1-> S )
Distinct variable groups:    g, a,  .+    G, a, g    g, H    X, a, g    .0. , a
Allowed substitution hints:    S( g, a)    F( g, a)    H( a)    .0. ( g)

Proof of Theorem cayleylem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq1 5562 . . . 4  |-  ( ( F `  x )  =  ( 0g `  H )  ->  (
( F `  x
) `  .0.  )  =  ( ( 0g
`  H ) `  .0.  ) )
2 simpr 447 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  x  e.  X )
3 cayleylem1.x . . . . . . . . 9  |-  X  =  ( Base `  G
)
4 cayleylem1.u . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
53, 4grpidcl 14559 . . . . . . . 8  |-  ( G  e.  Grp  ->  .0.  e.  X )
65adantr 451 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  .0.  e.  X )
7 cayleylem1.f . . . . . . . 8  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
87, 3grplactval 14612 . . . . . . 7  |-  ( ( x  e.  X  /\  .0.  e.  X )  -> 
( ( F `  x ) `  .0.  )  =  ( x  .+  .0.  ) )
92, 6, 8syl2anc 642 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x ) `  .0.  )  =  ( x  .+  .0.  ) )
10 cayleylem1.p . . . . . . 7  |-  .+  =  ( +g  `  G )
113, 10, 4grprid 14562 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x  .+  .0.  )  =  x )
129, 11eqtrd 2348 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x ) `  .0.  )  =  x )
13 fvex 5577 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
143, 13eqeltri 2386 . . . . . . . 8  |-  X  e. 
_V
15 cayleylem1.h . . . . . . . . 9  |-  H  =  ( SymGrp `  X )
1615symgid 14830 . . . . . . . 8  |-  ( X  e.  _V  ->  (  _I  |`  X )  =  ( 0g `  H
) )
1714, 16ax-mp 8 . . . . . . 7  |-  (  _I  |`  X )  =  ( 0g `  H )
1817fveq1i 5564 . . . . . 6  |-  ( (  _I  |`  X ) `  .0.  )  =  ( ( 0g `  H
) `  .0.  )
19 fvresi 5750 . . . . . . 7  |-  (  .0. 
e.  X  ->  (
(  _I  |`  X ) `
 .0.  )  =  .0.  )
206, 19syl 15 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( (  _I  |`  X ) `
 .0.  )  =  .0.  )
2118, 20syl5eqr 2362 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( 0g `  H ) `  .0.  )  =  .0.  )
2212, 21eqeq12d 2330 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( ( F `
 x ) `  .0.  )  =  (
( 0g `  H
) `  .0.  )  <->  x  =  .0.  ) )
231, 22syl5ib 210 . . 3  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x )  =  ( 0g `  H )  ->  x  =  .0.  ) )
2423ralrimiva 2660 . 2  |-  ( G  e.  Grp  ->  A. x  e.  X  ( ( F `  x )  =  ( 0g `  H )  ->  x  =  .0.  ) )
25 cayleylem1.s . . . 4  |-  S  =  ( Base `  H
)
263, 10, 4, 15, 25, 7cayleylem1 14836 . . 3  |-  ( G  e.  Grp  ->  F  e.  ( G  GrpHom  H ) )
27 eqid 2316 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
283, 25, 4, 27ghmf1 14760 . . 3  |-  ( F  e.  ( G  GrpHom  H )  ->  ( F : X -1-1-> S  <->  A. x  e.  X  ( ( F `  x )  =  ( 0g `  H )  ->  x  =  .0.  ) ) )
2926, 28syl 15 . 2  |-  ( G  e.  Grp  ->  ( F : X -1-1-> S  <->  A. x  e.  X  ( ( F `  x )  =  ( 0g `  H )  ->  x  =  .0.  ) ) )
3024, 29mpbird 223 1  |-  ( G  e.  Grp  ->  F : X -1-1-> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   _Vcvv 2822    e. cmpt 4114    _I cid 4341    |` cres 4728   -1-1->wf1 5289   ` cfv 5292  (class class class)co 5900   Basecbs 13195   +g cplusg 13255   0gc0g 13449   Grpcgrp 14411    GrpHom cghm 14729   SymGrpcsymg 14818
This theorem is referenced by:  cayley  14838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-tset 13274  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-sbg 14540  df-subg 14667  df-ghm 14730  df-ga 14793  df-symg 14819
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