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Theorem ga0 15065
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 4331 . . 3  |-  (/)  e.  _V
21jctr 527 . 2  |-  ( G  e.  Grp  ->  ( G  e.  Grp  /\  (/)  e.  _V ) )
3 f0 5619 . . . . 5  |-  (/) : (/) --> (/)
4 xp0 5283 . . . . . 6  |-  ( (
Base `  G )  X.  (/) )  =  (/)
54feq2i 5578 . . . . 5  |-  ( (/) : ( ( Base `  G
)  X.  (/) ) --> (/)  <->  (/) : (/) --> (/) )
63, 5mpbir 201 . . . 4  |-  (/) : ( ( Base `  G
)  X.  (/) ) --> (/)
7 ral0 3724 . . . 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 2435 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2435 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2435 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
1310, 11, 12isga 15058 . 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   (/)c0 3620    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13459   +g cplusg 13519   0gc0g 13713   Grpcgrp 14675    GrpAct cga 15056
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-ga 15057
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