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Theorem gaid 15031
Description: The trivial action of a group on any set. Each group element corresponds to the identity permutation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
gaid.1  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
gaid  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( 2nd  |`  ( X  X.  S ) )  e.  ( G  GrpAct  S ) )

Proof of Theorem gaid
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2924 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
21anim2i 553 . 2  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( G  e.  Grp  /\  S  e.  _V )
)
3 gaid.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
4 eqid 2404 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 14788 . . . . . . 7  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
65adantr 452 . . . . . 6  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( 0g `  G
)  e.  X )
7 ovres 6172 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  S )  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  ( ( 0g
`  G ) 2nd x ) )
8 df-ov 6043 . . . . . . . 8  |-  ( ( 0g `  G ) 2nd x )  =  ( 2nd `  <. ( 0g `  G ) ,  x >. )
9 fvex 5701 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
10 vex 2919 . . . . . . . . 9  |-  x  e. 
_V
119, 10op2nd 6315 . . . . . . . 8  |-  ( 2nd `  <. ( 0g `  G ) ,  x >. )  =  x
128, 11eqtri 2424 . . . . . . 7  |-  ( ( 0g `  G ) 2nd x )  =  x
137, 12syl6eq 2452 . . . . . 6  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  S )  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
146, 13sylan 458 . . . . 5  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x )
15 simprl 733 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  y  e.  X )
16 simplr 732 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  x  e.  S )
17 ovres 6172 . . . . . . . . 9  |-  ( ( y  e.  X  /\  x  e.  S )  ->  ( y ( 2nd  |`  ( X  X.  S
) ) x )  =  ( y 2nd x ) )
18 df-ov 6043 . . . . . . . . . 10  |-  ( y 2nd x )  =  ( 2nd `  <. y ,  x >. )
19 vex 2919 . . . . . . . . . . 11  |-  y  e. 
_V
2019, 10op2nd 6315 . . . . . . . . . 10  |-  ( 2nd `  <. y ,  x >. )  =  x
2118, 20eqtri 2424 . . . . . . . . 9  |-  ( y 2nd x )  =  x
2217, 21syl6eq 2452 . . . . . . . 8  |-  ( ( y  e.  X  /\  x  e.  S )  ->  ( y ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
2315, 16, 22syl2anc 643 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
( 2nd  |`  ( X  X.  S ) ) x )  =  x )
24 simprr 734 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  z  e.  X )
25 ovres 6172 . . . . . . . . . 10  |-  ( ( z  e.  X  /\  x  e.  S )  ->  ( z ( 2nd  |`  ( X  X.  S
) ) x )  =  ( z 2nd x ) )
26 df-ov 6043 . . . . . . . . . . 11  |-  ( z 2nd x )  =  ( 2nd `  <. z ,  x >. )
27 vex 2919 . . . . . . . . . . . 12  |-  z  e. 
_V
2827, 10op2nd 6315 . . . . . . . . . . 11  |-  ( 2nd `  <. z ,  x >. )  =  x
2926, 28eqtri 2424 . . . . . . . . . 10  |-  ( z 2nd x )  =  x
3025, 29syl6eq 2452 . . . . . . . . 9  |-  ( ( z  e.  X  /\  x  e.  S )  ->  ( z ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
3124, 16, 30syl2anc 643 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( z
( 2nd  |`  ( X  X.  S ) ) x )  =  x )
3231oveq2d 6056 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
( 2nd  |`  ( X  X.  S ) ) ( z ( 2nd  |`  ( X  X.  S
) ) x ) )  =  ( y ( 2nd  |`  ( X  X.  S ) ) x ) )
33 simpll 731 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  G  e.  Grp )
34 eqid 2404 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
353, 34grpcl 14773 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  z  e.  X )  ->  ( y ( +g  `  G ) z )  e.  X )
36353expb 1154 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( y  e.  X  /\  z  e.  X
) )  ->  (
y ( +g  `  G
) z )  e.  X )
3733, 36sylan 458 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
( +g  `  G ) z )  e.  X
)
38 ovres 6172 . . . . . . . . 9  |-  ( ( ( y ( +g  `  G ) z )  e.  X  /\  x  e.  S )  ->  (
( y ( +g  `  G ) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  ( ( y ( +g  `  G ) z ) 2nd x ) )
39 df-ov 6043 . . . . . . . . . 10  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  ( 2nd `  <. (
y ( +g  `  G
) z ) ,  x >. )
40 ovex 6065 . . . . . . . . . . 11  |-  ( y ( +g  `  G
) z )  e. 
_V
4140, 10op2nd 6315 . . . . . . . . . 10  |-  ( 2nd `  <. ( y ( +g  `  G ) z ) ,  x >. )  =  x
4239, 41eqtri 2424 . . . . . . . . 9  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  x
4338, 42syl6eq 2452 . . . . . . . 8  |-  ( ( ( y ( +g  `  G ) z )  e.  X  /\  x  e.  S )  ->  (
( y ( +g  `  G ) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x )
4437, 16, 43syl2anc 643 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( (
y ( +g  `  G
) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x )
4523, 32, 443eqtr4rd 2447 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( (
y ( +g  `  G
) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  ( y ( 2nd  |`  ( X  X.  S ) ) ( z ( 2nd  |`  ( X  X.  S
) ) x ) ) )
4645ralrimivva 2758 . . . . 5  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  A. y  e.  X  A. z  e.  X  ( (
y ( +g  `  G
) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  ( y ( 2nd  |`  ( X  X.  S ) ) ( z ( 2nd  |`  ( X  X.  S
) ) x ) ) )
4714, 46jca 519 . . . 4  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  ( (
( 0g `  G
) ( 2nd  |`  ( X  X.  S ) ) x )  =  x  /\  A. y  e.  X  A. z  e.  X  ( ( y ( +g  `  G
) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  ( y ( 2nd  |`  ( X  X.  S ) ) ( z ( 2nd  |`  ( X  X.  S
) ) x ) ) ) )
4847ralrimiva 2749 . . 3  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  A. x  e.  S  ( ( ( 0g
`  G ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x  /\  A. y  e.  X  A. z  e.  X  (
( y ( +g  `  G ) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  ( y ( 2nd  |`  ( X  X.  S ) ) ( z ( 2nd  |`  ( X  X.  S
) ) x ) ) ) )
49 f2ndres 6328 . . 3  |-  ( 2nd  |`  ( X  X.  S
) ) : ( X  X.  S ) --> S
5048, 49jctil 524 . 2  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( ( 2nd  |`  ( X  X.  S ) ) : ( X  X.  S ) --> S  /\  A. x  e.  S  ( ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  x  /\  A. y  e.  X  A. z  e.  X  (
( y ( +g  `  G ) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  ( y ( 2nd  |`  ( X  X.  S ) ) ( z ( 2nd  |`  ( X  X.  S
) ) x ) ) ) ) )
513, 34, 4isga 15023 . 2  |-  ( ( 2nd  |`  ( X  X.  S ) )  e.  ( G  GrpAct  S )  <-> 
( ( G  e. 
Grp  /\  S  e.  _V )  /\  (
( 2nd  |`  ( X  X.  S ) ) : ( X  X.  S ) --> S  /\  A. x  e.  S  ( ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  x  /\  A. y  e.  X  A. z  e.  X  (
( y ( +g  `  G ) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  ( y ( 2nd  |`  ( X  X.  S ) ) ( z ( 2nd  |`  ( X  X.  S
) ) x ) ) ) ) ) )
522, 50, 51sylanbrc 646 1  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( 2nd  |`  ( X  X.  S ) )  e.  ( G  GrpAct  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   <.cop 3777    X. cxp 4835    |` cres 4839   -->wf 5409   ` cfv 5413  (class class class)co 6040   2ndc2nd 6307   Basecbs 13424   +g cplusg 13484   0gc0g 13678   Grpcgrp 14640    GrpAct cga 15021
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-map 6979  df-0g 13682  df-mnd 14645  df-grp 14767  df-ga 15022
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