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Theorem gaid 15797
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 2971 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
21anim2i 564 . 2  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( G  e.  Grp  /\  S  e.  _V )
)
3 gaid.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
4 eqid 2433 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 15546 . . . . . . 7  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
65adantr 462 . . . . . 6  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( 0g `  G
)  e.  X )
7 ovres 6219 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  S )  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  ( ( 0g
`  G ) 2nd x ) )
8 df-ov 6083 . . . . . . . 8  |-  ( ( 0g `  G ) 2nd x )  =  ( 2nd `  <. ( 0g `  G ) ,  x >. )
9 fvex 5689 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
10 vex 2965 . . . . . . . . 9  |-  x  e. 
_V
119, 10op2nd 6575 . . . . . . . 8  |-  ( 2nd `  <. ( 0g `  G ) ,  x >. )  =  x
128, 11eqtri 2453 . . . . . . 7  |-  ( ( 0g `  G ) 2nd x )  =  x
137, 12syl6eq 2481 . . . . . 6  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  S )  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
146, 13sylan 468 . . . . 5  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x )
15 simprl 748 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  y  e.  X )
16 simplr 747 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  x  e.  S )
17 ovres 6219 . . . . . . . . 9  |-  ( ( y  e.  X  /\  x  e.  S )  ->  ( y ( 2nd  |`  ( X  X.  S
) ) x )  =  ( y 2nd x ) )
18 df-ov 6083 . . . . . . . . . 10  |-  ( y 2nd x )  =  ( 2nd `  <. y ,  x >. )
19 vex 2965 . . . . . . . . . . 11  |-  y  e. 
_V
2019, 10op2nd 6575 . . . . . . . . . 10  |-  ( 2nd `  <. y ,  x >. )  =  x
2118, 20eqtri 2453 . . . . . . . . 9  |-  ( y 2nd x )  =  x
2217, 21syl6eq 2481 . . . . . . . 8  |-  ( ( y  e.  X  /\  x  e.  S )  ->  ( y ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
2315, 16, 22syl2anc 654 . . . . . . 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 749 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  z  e.  X )
25 ovres 6219 . . . . . . . . . 10  |-  ( ( z  e.  X  /\  x  e.  S )  ->  ( z ( 2nd  |`  ( X  X.  S
) ) x )  =  ( z 2nd x ) )
26 df-ov 6083 . . . . . . . . . . 11  |-  ( z 2nd x )  =  ( 2nd `  <. z ,  x >. )
27 vex 2965 . . . . . . . . . . . 12  |-  z  e. 
_V
2827, 10op2nd 6575 . . . . . . . . . . 11  |-  ( 2nd `  <. z ,  x >. )  =  x
2926, 28eqtri 2453 . . . . . . . . . 10  |-  ( z 2nd x )  =  x
3025, 29syl6eq 2481 . . . . . . . . 9  |-  ( ( z  e.  X  /\  x  e.  S )  ->  ( z ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
3124, 16, 30syl2anc 654 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( z
( 2nd  |`  ( X  X.  S ) ) x )  =  x )
3231oveq2d 6096 . . . . . . 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 746 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  G  e.  Grp )
34 eqid 2433 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
353, 34grpcl 15531 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  z  e.  X )  ->  ( y ( +g  `  G ) z )  e.  X )
36353expb 1181 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( y  e.  X  /\  z  e.  X
) )  ->  (
y ( +g  `  G
) z )  e.  X )
3733, 36sylan 468 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
( +g  `  G ) z )  e.  X
)
38 ovres 6219 . . . . . . . . 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 6083 . . . . . . . . . 10  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  ( 2nd `  <. (
y ( +g  `  G
) z ) ,  x >. )
40 ovex 6105 . . . . . . . . . . 11  |-  ( y ( +g  `  G
) z )  e. 
_V
4140, 10op2nd 6575 . . . . . . . . . 10  |-  ( 2nd `  <. ( y ( +g  `  G ) z ) ,  x >. )  =  x
4239, 41eqtri 2453 . . . . . . . . 9  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  x
4338, 42syl6eq 2481 . . . . . . . 8  |-  ( ( ( y ( +g  `  G ) z )  e.  X  /\  x  e.  S )  ->  (
( y ( +g  `  G ) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x )
4437, 16, 43syl2anc 654 . . . . . . 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 2476 . . . . . 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 2798 . . . . 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 529 . . . 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 2789 . . 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 6588 . . 3  |-  ( 2nd  |`  ( X  X.  S
) ) : ( X  X.  S ) --> S
5048, 49jctil 534 . 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 15789 . 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 657 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 369    = wceq 1362    e. wcel 1755   A.wral 2705   _Vcvv 2962   <.cop 3871    X. cxp 4825    |` cres 4829   -->wf 5402   ` cfv 5406  (class class class)co 6080   2ndc2nd 6565   Basecbs 14157   +g cplusg 14221   0gc0g 14361   Grpcgrp 15393    GrpAct cga 15787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-2nd 6567  df-map 7204  df-0g 14363  df-mnd 15398  df-grp 15525  df-ga 15788
This theorem is referenced by: (None)
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