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Theorem ga0 16903
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 4557 . . 3  |-  (/)  e.  _V
21jctr 544 . 2  |-  ( G  e.  Grp  ->  ( G  e.  Grp  /\  (/)  e.  _V ) )
3 f0 5781 . . . . 5  |-  (/) : (/) --> (/)
4 xp0 5275 . . . . . 6  |-  ( (
Base `  G )  X.  (/) )  =  (/)
54feq2i 5739 . . . . 5  |-  ( (/) : ( ( Base `  G
)  X.  (/) ) --> (/)  <->  (/) : (/) --> (/) )
63, 5mpbir 212 . . . 4  |-  (/) : ( ( Base `  G
)  X.  (/) ) --> (/)
7 ral0 3908 . . . 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 456 . . 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 2429 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2429 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2429 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
1310, 11, 12isga 16896 . 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 668 1  |-  ( G  e.  Grp  ->  (/)  e.  ( G  GrpAct  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087   (/)c0 3767    X. cxp 4852   -->wf 5597   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   0gc0g 15297   Grpcgrp 16620    GrpAct cga 16894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-ga 16895
This theorem is referenced by: (None)
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