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Theorem gaid 15004
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 2908 . . 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 2388 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 14761 . . . . . . 7  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
65adantr 452 . . . . . 6  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( 0g `  G
)  e.  X )
7 ovres 6153 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  S )  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  ( ( 0g
`  G ) 2nd x ) )
8 df-ov 6024 . . . . . . . 8  |-  ( ( 0g `  G ) 2nd x )  =  ( 2nd `  <. ( 0g `  G ) ,  x >. )
9 fvex 5683 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
10 vex 2903 . . . . . . . . 9  |-  x  e. 
_V
119, 10op2nd 6296 . . . . . . . 8  |-  ( 2nd `  <. ( 0g `  G ) ,  x >. )  =  x
128, 11eqtri 2408 . . . . . . 7  |-  ( ( 0g `  G ) 2nd x )  =  x
137, 12syl6eq 2436 . . . . . 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 6153 . . . . . . . . 9  |-  ( ( y  e.  X  /\  x  e.  S )  ->  ( y ( 2nd  |`  ( X  X.  S
) ) x )  =  ( y 2nd x ) )
18 df-ov 6024 . . . . . . . . . 10  |-  ( y 2nd x )  =  ( 2nd `  <. y ,  x >. )
19 vex 2903 . . . . . . . . . . 11  |-  y  e. 
_V
2019, 10op2nd 6296 . . . . . . . . . 10  |-  ( 2nd `  <. y ,  x >. )  =  x
2118, 20eqtri 2408 . . . . . . . . 9  |-  ( y 2nd x )  =  x
2217, 21syl6eq 2436 . . . . . . . 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 6153 . . . . . . . . . 10  |-  ( ( z  e.  X  /\  x  e.  S )  ->  ( z ( 2nd  |`  ( X  X.  S
) ) x )  =  ( z 2nd x ) )
26 df-ov 6024 . . . . . . . . . . 11  |-  ( z 2nd x )  =  ( 2nd `  <. z ,  x >. )
27 vex 2903 . . . . . . . . . . . 12  |-  z  e. 
_V
2827, 10op2nd 6296 . . . . . . . . . . 11  |-  ( 2nd `  <. z ,  x >. )  =  x
2926, 28eqtri 2408 . . . . . . . . . 10  |-  ( z 2nd x )  =  x
3025, 29syl6eq 2436 . . . . . . . . 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 6037 . . . . . . 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 2388 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
353, 34grpcl 14746 . . . . . . . . . 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 6153 . . . . . . . . 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 6024 . . . . . . . . . 10  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  ( 2nd `  <. (
y ( +g  `  G
) z ) ,  x >. )
40 ovex 6046 . . . . . . . . . . 11  |-  ( y ( +g  `  G
) z )  e. 
_V
4140, 10op2nd 6296 . . . . . . . . . 10  |-  ( 2nd `  <. ( y ( +g  `  G ) z ) ,  x >. )  =  x
4239, 41eqtri 2408 . . . . . . . . 9  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  x
4338, 42syl6eq 2436 . . . . . . . 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 2431 . . . . . 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 2742 . . . . 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 2733 . . 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 6309 . . 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 14996 . 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 1717   A.wral 2650   _Vcvv 2900   <.cop 3761    X. cxp 4817    |` cres 4821   -->wf 5391   ` cfv 5395  (class class class)co 6021   2ndc2nd 6288   Basecbs 13397   +g cplusg 13457   0gc0g 13651   Grpcgrp 14613    GrpAct cga 14994
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-map 6957  df-0g 13655  df-mnd 14618  df-grp 14740  df-ga 14995
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