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Theorem orbstafun 14814
Description: Existence and uniqueness for the function of orbsta 14816. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
gasta.1  |-  X  =  ( Base `  G
)
gasta.2  |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }
orbsta.r  |-  .~  =  ( G ~QG  H )
orbsta.f  |-  F  =  ran  ( k  e.  X  |->  <. [ k ]  .~  ,  ( k 
.(+)  A ) >. )
Assertion
Ref Expression
orbstafun  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  Fun  F )
Distinct variable groups:    .~ , k    u, k,  .(+)    A, k, u    k, G, u    k, X, u   
k, Y
Allowed substitution hints:    .~ ( u)    F( u, k)    H( u, k)    Y( u)

Proof of Theorem orbstafun
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 orbsta.f . 2  |-  F  =  ran  ( k  e.  X  |->  <. [ k ]  .~  ,  ( k 
.(+)  A ) >. )
2 ovex 5925 . . 3  |-  ( k 
.(+)  A )  e.  _V
32a1i 10 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  e.  X
)  ->  ( k  .(+)  A )  e.  _V )
4 gasta.1 . . . 4  |-  X  =  ( Base `  G
)
5 gasta.2 . . . 4  |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }
64, 5gastacl 14812 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  H  e.  (SubGrp `  G )
)
7 orbsta.r . . . 4  |-  .~  =  ( G ~QG  H )
84, 7eqger 14716 . . 3  |-  ( H  e.  (SubGrp `  G
)  ->  .~  Er  X
)
96, 8syl 15 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  .~  Er  X )
10 fvex 5577 . . . 4  |-  ( Base `  G )  e.  _V
114, 10eqeltri 2386 . . 3  |-  X  e. 
_V
1211a1i 10 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  X  e.  _V )
13 oveq1 5907 . 2  |-  ( k  =  h  ->  (
k  .(+)  A )  =  ( h  .(+)  A ) )
14 simpr 447 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  k  .~  h )
15 subgrcl 14675 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  G  e.  Grp )
164subgss 14671 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
17 eqid 2316 . . . . . . . . . 10  |-  ( inv g `  G )  =  ( inv g `  G )
18 eqid 2316 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
194, 17, 18, 7eqgval 14715 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  H  C_  X )  -> 
( k  .~  h  <->  ( k  e.  X  /\  h  e.  X  /\  ( ( ( inv g `  G ) `
 k ) ( +g  `  G ) h )  e.  H
) ) )
2015, 16, 19syl2anc 642 . . . . . . . 8  |-  ( H  e.  (SubGrp `  G
)  ->  ( k  .~  h  <->  ( k  e.  X  /\  h  e.  X  /\  ( ( ( inv g `  G ) `  k
) ( +g  `  G
) h )  e.  H ) ) )
216, 20syl 15 . . . . . . 7  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  (
k  .~  h  <->  ( k  e.  X  /\  h  e.  X  /\  (
( ( inv g `  G ) `  k
) ( +g  `  G
) h )  e.  H ) ) )
2221biimpa 470 . . . . . 6  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  ( k  e.  X  /\  h  e.  X  /\  (
( ( inv g `  G ) `  k
) ( +g  `  G
) h )  e.  H ) )
2322simp1d 967 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  k  e.  X )
2422simp2d 968 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  h  e.  X )
2523, 24jca 518 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  ( k  e.  X  /\  h  e.  X ) )
264, 5, 7gastacos 14813 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  ( k  e.  X  /\  h  e.  X
) )  ->  (
k  .~  h  <->  ( k  .(+)  A )  =  ( h  .(+)  A )
) )
2725, 26syldan 456 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  ( k  .~  h  <->  ( k  .(+)  A )  =  ( h 
.(+)  A ) ) )
2814, 27mpbid 201 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  ( k  .(+)  A )  =  ( h  .(+)  A )
)
291, 3, 9, 12, 13, 28qliftfund 6787 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  Fun  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   {crab 2581   _Vcvv 2822    C_ wss 3186   <.cop 3677   class class class wbr 4060    e. cmpt 4114   ran crn 4727   Fun wfun 5286   ` cfv 5292  (class class class)co 5900    Er wer 6699   [cec 6700   Basecbs 13195   +g cplusg 13255   Grpcgrp 14411   inv gcminusg 14412  SubGrpcsubg 14664   ~QG cqg 14666    GrpAct cga 14792
This theorem is referenced by:  orbstaval  14815  orbsta  14816
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-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-er 6702  df-ec 6704  df-qs 6708  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  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-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-subg 14667  df-eqg 14669  df-ga 14793
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