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Theorem cayleylem2 16038
Description: Lemma for cayley 16039. (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 5799 . . . 4  |-  ( ( F `  x )  =  ( 0g `  H )  ->  (
( F `  x
) `  .0.  )  =  ( ( 0g
`  H ) `  .0.  ) )
2 simpr 461 . . . . . . 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 15686 . . . . . . . 8  |-  ( G  e.  Grp  ->  .0.  e.  X )
65adantr 465 . . . . . . 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 15743 . . . . . . 7  |-  ( ( x  e.  X  /\  .0.  e.  X )  -> 
( ( F `  x ) `  .0.  )  =  ( x  .+  .0.  ) )
92, 6, 8syl2anc 661 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x ) `  .0.  )  =  ( x  .+  .0.  ) )
10 cayleylem1.p . . . . . . 7  |-  .+  =  ( +g  `  G )
113, 10, 4grprid 15689 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x  .+  .0.  )  =  x )
129, 11eqtrd 2495 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x ) `  .0.  )  =  x )
13 fvex 5810 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
143, 13eqeltri 2538 . . . . . . . 8  |-  X  e. 
_V
15 cayleylem1.h . . . . . . . . 9  |-  H  =  ( SymGrp `  X )
1615symgid 16026 . . . . . . . 8  |-  ( X  e.  _V  ->  (  _I  |`  X )  =  ( 0g `  H
) )
1714, 16ax-mp 5 . . . . . . 7  |-  (  _I  |`  X )  =  ( 0g `  H )
1817fveq1i 5801 . . . . . 6  |-  ( (  _I  |`  X ) `  .0.  )  =  ( ( 0g `  H
) `  .0.  )
19 fvresi 6014 . . . . . . 7  |-  (  .0. 
e.  X  ->  (
(  _I  |`  X ) `
 .0.  )  =  .0.  )
206, 19syl 16 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( (  _I  |`  X ) `
 .0.  )  =  .0.  )
2118, 20syl5eqr 2509 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( 0g `  H ) `  .0.  )  =  .0.  )
2212, 21eqeq12d 2476 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( ( F `
 x ) `  .0.  )  =  (
( 0g `  H
) `  .0.  )  <->  x  =  .0.  ) )
231, 22syl5ib 219 . . 3  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x )  =  ( 0g `  H )  ->  x  =  .0.  ) )
2423ralrimiva 2830 . 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 16037 . . 3  |-  ( G  e.  Grp  ->  F  e.  ( G  GrpHom  H ) )
27 eqid 2454 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
283, 25, 4, 27ghmf1 15895 . . 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 232 1  |-  ( G  e.  Grp  ->  F : X -1-1-> S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    |-> cmpt 4459    _I cid 4740    |` cres 4951   -1-1->wf1 5524   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   0gc0g 14498   Grpcgrp 15530    GrpHom cghm 15864   SymGrpcsymg 16002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-tset 14377  df-0g 14500  df-mnd 15535  df-grp 15665  df-minusg 15666  df-sbg 15667  df-subg 15798  df-ghm 15865  df-ga 15928  df-symg 16003
This theorem is referenced by:  cayley  16039
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