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Theorem ga0 15004
Description: The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Assertion
Ref Expression
ga0  |-  ( G  e.  Grp  ->  (/)  e.  ( G  GrpAct  (/) ) )

Proof of Theorem ga0
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4282 . . 3  |-  (/)  e.  _V
21jctr 527 . 2  |-  ( G  e.  Grp  ->  ( G  e.  Grp  /\  (/)  e.  _V ) )
3 f0 5569 . . . . 5  |-  (/) : (/) --> (/)
4 xp0 5233 . . . . . 6  |-  ( (
Base `  G )  X.  (/) )  =  (/)
54feq2i 5528 . . . . 5  |-  ( (/) : ( ( Base `  G
)  X.  (/) ) --> (/)  <->  (/) : (/) --> (/) )
63, 5mpbir 201 . . . 4  |-  (/) : ( ( Base `  G
)  X.  (/) ) --> (/)
7 ral0 3677 . . . 4  |-  A. x  e.  (/)  ( ( ( 0g `  G )
(/) x )  =  x  /\  A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z ) (/) x )  =  ( y (/) ( z (/) x ) ) )
86, 7pm3.2i 442 . . 3  |-  ( (/) : ( ( Base `  G
)  X.  (/) ) --> (/)  /\ 
A. x  e.  (/)  ( ( ( 0g
`  G ) (/) x )  =  x  /\  A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z ) (/) x )  =  ( y (/) ( z (/) x ) ) ) )
98a1i 11 . 2  |-  ( G  e.  Grp  ->  ( (/)
: ( ( Base `  G )  X.  (/) ) --> (/)  /\ 
A. x  e.  (/)  ( ( ( 0g
`  G ) (/) x )  =  x  /\  A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z ) (/) x )  =  ( y (/) ( z (/) x ) ) ) ) )
10 eqid 2389 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2389 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2389 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
1310, 11, 12isga 14997 . 2  |-  ( (/)  e.  ( G  GrpAct  (/) )  <->  ( ( G  e.  Grp  /\  (/)  e.  _V )  /\  ( (/) : ( ( Base `  G
)  X.  (/) ) --> (/)  /\ 
A. x  e.  (/)  ( ( ( 0g
`  G ) (/) x )  =  x  /\  A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z ) (/) x )  =  ( y (/) ( z (/) x ) ) ) ) ) )
142, 9, 13sylanbrc 646 1  |-  ( G  e.  Grp  ->  (/)  e.  ( G  GrpAct  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   _Vcvv 2901   (/)c0 3573    X. cxp 4818   -->wf 5392   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458   0gc0g 13652   Grpcgrp 14614    GrpAct cga 14995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-map 6958  df-ga 14996
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