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Theorem cayleylem2 15111
Description: Lemma for cayley 15112. (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 5727 . . . 4  |-  ( ( F `  x )  =  ( 0g `  H )  ->  (
( F `  x
) `  .0.  )  =  ( ( 0g
`  H ) `  .0.  ) )
2 simpr 448 . . . . . . 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 14833 . . . . . . . 8  |-  ( G  e.  Grp  ->  .0.  e.  X )
65adantr 452 . . . . . . 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 14886 . . . . . . 7  |-  ( ( x  e.  X  /\  .0.  e.  X )  -> 
( ( F `  x ) `  .0.  )  =  ( x  .+  .0.  ) )
92, 6, 8syl2anc 643 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x ) `  .0.  )  =  ( x  .+  .0.  ) )
10 cayleylem1.p . . . . . . 7  |-  .+  =  ( +g  `  G )
113, 10, 4grprid 14836 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x  .+  .0.  )  =  x )
129, 11eqtrd 2468 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x ) `  .0.  )  =  x )
13 fvex 5742 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
143, 13eqeltri 2506 . . . . . . . 8  |-  X  e. 
_V
15 cayleylem1.h . . . . . . . . 9  |-  H  =  ( SymGrp `  X )
1615symgid 15104 . . . . . . . 8  |-  ( X  e.  _V  ->  (  _I  |`  X )  =  ( 0g `  H
) )
1714, 16ax-mp 8 . . . . . . 7  |-  (  _I  |`  X )  =  ( 0g `  H )
1817fveq1i 5729 . . . . . 6  |-  ( (  _I  |`  X ) `  .0.  )  =  ( ( 0g `  H
) `  .0.  )
19 fvresi 5924 . . . . . . 7  |-  (  .0. 
e.  X  ->  (
(  _I  |`  X ) `
 .0.  )  =  .0.  )
206, 19syl 16 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( (  _I  |`  X ) `
 .0.  )  =  .0.  )
2118, 20syl5eqr 2482 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( 0g `  H ) `  .0.  )  =  .0.  )
2212, 21eqeq12d 2450 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( ( F `
 x ) `  .0.  )  =  (
( 0g `  H
) `  .0.  )  <->  x  =  .0.  ) )
231, 22syl5ib 211 . . 3  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x )  =  ( 0g `  H )  ->  x  =  .0.  ) )
2423ralrimiva 2789 . 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 15110 . . 3  |-  ( G  e.  Grp  ->  F  e.  ( G  GrpHom  H ) )
27 eqid 2436 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
283, 25, 4, 27ghmf1 15034 . . 3  |-  ( F  e.  ( G  GrpHom  H )  ->  ( F : X -1-1-> S  <->  A. x  e.  X  ( ( F `  x )  =  ( 0g `  H )  ->  x  =  .0.  ) ) )
2926, 28syl 16 . 2  |-  ( G  e.  Grp  ->  ( F : X -1-1-> S  <->  A. x  e.  X  ( ( F `  x )  =  ( 0g `  H )  ->  x  =  .0.  ) ) )
3024, 29mpbird 224 1  |-  ( G  e.  Grp  ->  F : X -1-1-> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    e. cmpt 4266    _I cid 4493    |` cres 4880   -1-1->wf1 5451   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685    GrpHom cghm 15003   SymGrpcsymg 15092
This theorem is referenced by:  cayley  15112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-tset 13548  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-ghm 15004  df-ga 15067  df-symg 15093
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