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Theorem orbstafun 15088
Description: Existence and uniqueness for the function of orbsta 15090. (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 6106 . . 3  |-  ( k 
.(+)  A )  e.  _V
32a1i 11 . 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 15086 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  H  e.  (SubGrp `  G )
)
7 orbsta.r . . . 4  |-  .~  =  ( G ~QG  H )
84, 7eqger 14990 . . 3  |-  ( H  e.  (SubGrp `  G
)  ->  .~  Er  X
)
96, 8syl 16 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  .~  Er  X )
10 fvex 5742 . . . 4  |-  ( Base `  G )  e.  _V
114, 10eqeltri 2506 . . 3  |-  X  e. 
_V
1211a1i 11 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  X  e.  _V )
13 oveq1 6088 . 2  |-  ( k  =  h  ->  (
k  .(+)  A )  =  ( h  .(+)  A ) )
14 simpr 448 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  k  .~  h )
15 subgrcl 14949 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  G  e.  Grp )
164subgss 14945 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
17 eqid 2436 . . . . . . . . . 10  |-  ( inv g `  G )  =  ( inv g `  G )
18 eqid 2436 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
194, 17, 18, 7eqgval 14989 . . . . . . . . 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 643 . . . . . . . 8  |-  ( H  e.  (SubGrp `  G
)  ->  ( k  .~  h  <->  ( k  e.  X  /\  h  e.  X  /\  ( ( ( inv g `  G ) `  k
) ( +g  `  G
) h )  e.  H ) ) )
216, 20syl 16 . . . . . . 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 471 . . . . . 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 969 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  k  e.  X )
2422simp2d 970 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  h  e.  X )
2523, 24jca 519 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  ( k  e.  X  /\  h  e.  X ) )
264, 5, 7gastacos 15087 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  ( k  e.  X  /\  h  e.  X
) )  ->  (
k  .~  h  <->  ( k  .(+)  A )  =  ( h  .(+)  A )
) )
2725, 26syldan 457 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  ( k  .~  h  <->  ( k  .(+)  A )  =  ( h 
.(+)  A ) ) )
2814, 27mpbid 202 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  k  .~  h
)  ->  ( k  .(+)  A )  =  ( h  .(+)  A )
)
291, 3, 9, 12, 13, 28qliftfund 6990 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  Fun  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    C_ wss 3320   <.cop 3817   class class class wbr 4212    e. cmpt 4266   ran crn 4879   Fun wfun 5448   ` cfv 5454  (class class class)co 6081    Er wer 6902   [cec 6903   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685   inv gcminusg 14686  SubGrpcsubg 14938   ~QG cqg 14940    GrpAct cga 15066
This theorem is referenced by:  orbstaval  15089  orbsta  15090
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-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-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  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-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941  df-eqg 14943  df-ga 15067
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