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Theorem gaid 15940
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 3087 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
21anim2i 569 . 2  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( G  e.  Grp  /\  S  e.  _V )
)
3 gaid.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
4 eqid 2454 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 15689 . . . . . . 7  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
65adantr 465 . . . . . 6  |-  ( ( G  e.  Grp  /\  S  e.  V )  ->  ( 0g `  G
)  e.  X )
7 ovres 6343 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  S )  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  ( ( 0g
`  G ) 2nd x ) )
8 df-ov 6206 . . . . . . . 8  |-  ( ( 0g `  G ) 2nd x )  =  ( 2nd `  <. ( 0g `  G ) ,  x >. )
9 fvex 5812 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
10 vex 3081 . . . . . . . . 9  |-  x  e. 
_V
119, 10op2nd 6699 . . . . . . . 8  |-  ( 2nd `  <. ( 0g `  G ) ,  x >. )  =  x
128, 11eqtri 2483 . . . . . . 7  |-  ( ( 0g `  G ) 2nd x )  =  x
137, 12syl6eq 2511 . . . . . 6  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  S )  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
146, 13sylan 471 . . . . 5  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  ( ( 0g `  G ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x )
15 simprl 755 . . . . . . . 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 6343 . . . . . . . . 9  |-  ( ( y  e.  X  /\  x  e.  S )  ->  ( y ( 2nd  |`  ( X  X.  S
) ) x )  =  ( y 2nd x ) )
18 df-ov 6206 . . . . . . . . . 10  |-  ( y 2nd x )  =  ( 2nd `  <. y ,  x >. )
19 vex 3081 . . . . . . . . . . 11  |-  y  e. 
_V
2019, 10op2nd 6699 . . . . . . . . . 10  |-  ( 2nd `  <. y ,  x >. )  =  x
2118, 20eqtri 2483 . . . . . . . . 9  |-  ( y 2nd x )  =  x
2217, 21syl6eq 2511 . . . . . . . 8  |-  ( ( y  e.  X  /\  x  e.  S )  ->  ( y ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
2315, 16, 22syl2anc 661 . . . . . . 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 756 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  z  e.  X )
25 ovres 6343 . . . . . . . . . 10  |-  ( ( z  e.  X  /\  x  e.  S )  ->  ( z ( 2nd  |`  ( X  X.  S
) ) x )  =  ( z 2nd x ) )
26 df-ov 6206 . . . . . . . . . . 11  |-  ( z 2nd x )  =  ( 2nd `  <. z ,  x >. )
27 vex 3081 . . . . . . . . . . . 12  |-  z  e. 
_V
2827, 10op2nd 6699 . . . . . . . . . . 11  |-  ( 2nd `  <. z ,  x >. )  =  x
2926, 28eqtri 2483 . . . . . . . . . 10  |-  ( z 2nd x )  =  x
3025, 29syl6eq 2511 . . . . . . . . 9  |-  ( ( z  e.  X  /\  x  e.  S )  ->  ( z ( 2nd  |`  ( X  X.  S
) ) x )  =  x )
3124, 16, 30syl2anc 661 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( z
( 2nd  |`  ( X  X.  S ) ) x )  =  x )
3231oveq2d 6219 . . . . . . 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 753 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  S  e.  V )  /\  x  e.  S
)  ->  G  e.  Grp )
34 eqid 2454 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
353, 34grpcl 15674 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  z  e.  X )  ->  ( y ( +g  `  G ) z )  e.  X )
36353expb 1189 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( y  e.  X  /\  z  e.  X
) )  ->  (
y ( +g  `  G
) z )  e.  X )
3733, 36sylan 471 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  S  e.  V )  /\  x  e.  S )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
( +g  `  G ) z )  e.  X
)
38 ovres 6343 . . . . . . . . 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 6206 . . . . . . . . . 10  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  ( 2nd `  <. (
y ( +g  `  G
) z ) ,  x >. )
40 ovex 6228 . . . . . . . . . . 11  |-  ( y ( +g  `  G
) z )  e. 
_V
4140, 10op2nd 6699 . . . . . . . . . 10  |-  ( 2nd `  <. ( y ( +g  `  G ) z ) ,  x >. )  =  x
4239, 41eqtri 2483 . . . . . . . . 9  |-  ( ( y ( +g  `  G
) z ) 2nd x )  =  x
4338, 42syl6eq 2511 . . . . . . . 8  |-  ( ( ( y ( +g  `  G ) z )  e.  X  /\  x  e.  S )  ->  (
( y ( +g  `  G ) z ) ( 2nd  |`  ( X  X.  S ) ) x )  =  x )
4437, 16, 43syl2anc 661 . . . . . . 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 2506 . . . . . 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 2914 . . . . 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 532 . . . 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 2830 . . 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 6712 . . 3  |-  ( 2nd  |`  ( X  X.  S
) ) : ( X  X.  S ) --> S
5048, 49jctil 537 . 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 15932 . 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 664 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 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078   <.cop 3994    X. cxp 4949    |` cres 4953   -->wf 5525   ` cfv 5529  (class class class)co 6203   2ndc2nd 6689   Basecbs 14296   +g cplusg 14361   0gc0g 14501   Grpcgrp 15533    GrpAct cga 15930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-2nd 6691  df-map 7329  df-0g 14503  df-mnd 15538  df-grp 15668  df-ga 15931
This theorem is referenced by: (None)
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