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Theorem gapm2 24781
Description: The action of a particular group element is a permutation of the base set. gapm 14760 expressed with the currying operator. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 3-May-2015.)
Hypothesis
Ref Expression
gapm2.1  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
gapm2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  -> 
( ( cur1 `  .(+)  ) `  A ) : Y -1-1-onto-> Y
)

Proof of Theorem gapm2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 gapm2.1 . . . 4  |-  X  =  ( Base `  G
)
2 eqid 2283 . . . 4  |-  ( x  e.  Y  |->  ( A 
.(+)  x ) )  =  ( x  e.  Y  |->  ( A  .(+)  x ) )
31, 2gapm 14760 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  ->  (
x  e.  Y  |->  ( A  .(+)  x )
) : Y -1-1-onto-> Y )
43adantrr 697 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  -> 
( x  e.  Y  |->  ( A  .(+)  x ) ) : Y -1-1-onto-> Y )
5 fvex 5539 . . . . . 6  |-  ( Base `  G )  e.  _V
61, 5eqeltri 2353 . . . . 5  |-  X  e. 
_V
76a1i 10 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  X  e.  _V )
8 gaset 14747 . . . . 5  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  Y  e.  _V )
98adantr 451 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  Y  e.  _V )
10 simprr 733 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  Y  =/=  (/) )
111gaf 14749 . . . . . 6  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
1211adantr 451 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  .(+)  : ( X  X.  Y ) --> Y )
13 ffn 5389 . . . . 5  |-  (  .(+)  : ( X  X.  Y
) --> Y  ->  .(+)  Fn  ( X  X.  Y ) )
1412, 13syl 15 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  .(+)  Fn  ( X  X.  Y ) )
15 simprl 732 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  A  e.  X )
16 valcurfn2 24617 . . . 4  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( Y  =/=  (/)  /\  .(+)  Fn  ( X  X.  Y
) )  /\  A  e.  X )  ->  (
( cur1 `  .(+)  ) `  A )  =  ( x  e.  Y  |->  ( A  .(+)  x )
) )
177, 9, 10, 14, 15, 16syl221anc 1193 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  -> 
( ( cur1 `  .(+)  ) `  A )  =  ( x  e.  Y  |->  ( A  .(+)  x )
) )
18 f1oeq1 5463 . . 3  |-  ( ( ( cur1 `  .(+)  ) `  A )  =  ( x  e.  Y  |->  ( A  .(+)  x )
)  ->  ( (
( cur1 `  .(+)  ) `  A ) : Y -1-1-onto-> Y  <->  ( x  e.  Y  |->  ( A  .(+)  x )
) : Y -1-1-onto-> Y ) )
1917, 18syl 15 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  -> 
( ( ( cur1 `  .(+)  ) `  A
) : Y -1-1-onto-> Y  <->  ( x  e.  Y  |->  ( A 
.(+)  x ) ) : Y -1-1-onto-> Y ) )
204, 19mpbird 223 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  -> 
( ( cur1 `  .(+)  ) `  A ) : Y -1-1-onto-> Y
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148    GrpAct cga 14743   cur1ccur1 24606
This theorem is referenced by:  rngapm  24782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-ga 14744  df-cur1 24608
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