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Theorem cayleylem2 16640
Description: Lemma for cayley 16641. (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 5847 . . . 4  |-  ( ( F `  x )  =  ( 0g `  H )  ->  (
( F `  x
) `  .0.  )  =  ( ( 0g
`  H ) `  .0.  ) )
2 simpr 459 . . . . . . 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 16280 . . . . . . . 8  |-  ( G  e.  Grp  ->  .0.  e.  X )
65adantr 463 . . . . . . 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 16339 . . . . . . 7  |-  ( ( x  e.  X  /\  .0.  e.  X )  -> 
( ( F `  x ) `  .0.  )  =  ( x  .+  .0.  ) )
92, 6, 8syl2anc 659 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( F `  x ) `  .0.  )  =  ( x  .+  .0.  ) )
10 cayleylem1.p . . . . . . 7  |-  .+  =  ( +g  `  G )
113, 10, 4grprid 16283 . . . . . 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 5858 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
143, 13eqeltri 2538 . . . . . . . 8  |-  X  e. 
_V
15 cayleylem1.h . . . . . . . . 9  |-  H  =  ( SymGrp `  X )
1615symgid 16628 . . . . . . . 8  |-  ( X  e.  _V  ->  (  _I  |`  X )  =  ( 0g `  H
) )
1714, 16ax-mp 5 . . . . . . 7  |-  (  _I  |`  X )  =  ( 0g `  H )
1817fveq1i 5849 . . . . . 6  |-  ( (  _I  |`  X ) `  .0.  )  =  ( ( 0g `  H
) `  .0.  )
19 fvresi 6073 . . . . . . 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 2868 . 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 16639 . . 3  |-  ( G  e.  Grp  ->  F  e.  ( G  GrpHom  H ) )
27 eqid 2454 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
283, 25, 4, 27ghmf1 16497 . . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    |-> cmpt 4497    _I cid 4779    |` cres 4990   -1-1->wf1 5567   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   0gc0g 14932   Grpcgrp 16255    GrpHom cghm 16466   SymGrpcsymg 16604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-tset 14806  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-minusg 16260  df-sbg 16261  df-subg 16400  df-ghm 16467  df-ga 16530  df-symg 16605
This theorem is referenced by:  cayley  16641
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