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Theorem gaid 16659
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 3067 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
21anim2i 567 . 2  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( G  e.  Grp  /\  S  e.  _V )
)
3 gaid.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
4 eqid 2402 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 16400 . . . . . . 7  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
65adantr 463 . . . . . 6  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( 0g `  G
)  e.  X )
7 ovres 6422 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  S )  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  ( ( 0g
`  G ) 2nd x ) )
8 df-ov 6280 . . . . . . . 8  |-  ( ( 0g `  G ) 2nd x )  =  ( 2nd `  <. ( 0g `  G ) ,  x >. )
9 fvex 5858 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
10 vex 3061 . . . . . . . . 9  |-  x  e. 
_V
119, 10op2nd 6792 . . . . . . . 8  |-  ( 2nd `  <. ( 0g `  G ) ,  x >. )  =  x
128, 11eqtri 2431 . . . . . . 7  |-  ( ( 0g `  G ) 2nd x )  =  x
137, 12syl6eq 2459 . . . . . 6  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  S )  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
146, 13sylan 469 . . . . 5  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x )
15 simprl 756 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  y  e.  X )
16 simplr 754 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  x  e.  S )
17 ovres 6422 . . . . . . . . 9  |-  ( ( y  e.  X  /\  x  e.  S )  ->  ( y ( 2nd  |`  ( X  X.  S
) ) x )  =  ( y 2nd x ) )
18 df-ov 6280 . . . . . . . . . 10  |-  ( y 2nd x )  =  ( 2nd `  <. y ,  x >. )
19 vex 3061 . . . . . . . . . . 11  |-  y  e. 
_V
2019, 10op2nd 6792 . . . . . . . . . 10  |-  ( 2nd `  <. y ,  x >. )  =  x
2118, 20eqtri 2431 . . . . . . . . 9  |-  ( y 2nd x )  =  x
2217, 21syl6eq 2459 . . . . . . . 8  |-  ( ( y  e.  X  /\  x  e.  S )  ->  ( y ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
2315, 16, 22syl2anc 659 . . . . . . 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 758 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  z  e.  X )
25 ovres 6422 . . . . . . . . . 10  |-  ( ( z  e.  X  /\  x  e.  S )  ->  ( z ( 2nd  |`  ( X  X.  S
) ) x )  =  ( z 2nd x ) )
26 df-ov 6280 . . . . . . . . . . 11  |-  ( z 2nd x )  =  ( 2nd `  <. z ,  x >. )
27 vex 3061 . . . . . . . . . . . 12  |-  z  e. 
_V
2827, 10op2nd 6792 . . . . . . . . . . 11  |-  ( 2nd `  <. z ,  x >. )  =  x
2926, 28eqtri 2431 . . . . . . . . . 10  |-  ( z 2nd x )  =  x
3025, 29syl6eq 2459 . . . . . . . . 9  |-  ( ( z  e.  X  /\  x  e.  S )  ->  ( z ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
3124, 16, 30syl2anc 659 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( z
( 2nd  |`  ( X  X.  S ) ) x )  =  x )
3231oveq2d 6293 . . . . . . 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 752 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  G  e.  Grp )
34 eqid 2402 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
353, 34grpcl 16385 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  z  e.  X )  ->  ( y ( +g  `  G ) z )  e.  X )
36353expb 1198 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( y  e.  X  /\  z  e.  X
) )  ->  (
y ( +g  `  G
) z )  e.  X )
3733, 36sylan 469 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
( +g  `  G ) z )  e.  X
)
38 ovres 6422 . . . . . . . . 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 6280 . . . . . . . . . 10  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  ( 2nd `  <. (
y ( +g  `  G
) z ) ,  x >. )
40 ovex 6305 . . . . . . . . . . 11  |-  ( y ( +g  `  G
) z )  e. 
_V
4140, 10op2nd 6792 . . . . . . . . . 10  |-  ( 2nd `  <. ( y ( +g  `  G ) z ) ,  x >. )  =  x
4239, 41eqtri 2431 . . . . . . . . 9  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  x
4338, 42syl6eq 2459 . . . . . . . 8  |-  ( ( ( y ( +g  `  G ) z )  e.  X  /\  x  e.  S )  ->  (
( y ( +g  `  G ) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x )
4437, 16, 43syl2anc 659 . . . . . . 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 2454 . . . . . 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 2824 . . . . 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 530 . . . 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 2817 . . 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 6806 . . 3  |-  ( 2nd  |`  ( X  X.  S
) ) : ( X  X.  S ) --> S
5048, 49jctil 535 . 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 16651 . 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 662 1  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( 2nd  |`  ( X  X.  S ) )  e.  ( G  GrpAct  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   _Vcvv 3058   <.cop 3977    X. cxp 4820    |` cres 4824   -->wf 5564   ` cfv 5568  (class class class)co 6277   2ndc2nd 6782   Basecbs 14839   +g cplusg 14907   0gc0g 15052   Grpcgrp 16375    GrpAct cga 16649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-2nd 6784  df-map 7458  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-ga 16650
This theorem is referenced by: (None)
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