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Theorem subgga 16212
Description: A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
subgga.1  |-  X  =  ( Base `  G
)
subgga.2  |-  .+  =  ( +g  `  G )
subgga.3  |-  H  =  ( Gs  Y )
subgga.4  |-  F  =  ( x  e.  Y ,  y  e.  X  |->  ( x  .+  y
) )
Assertion
Ref Expression
subgga  |-  ( Y  e.  (SubGrp `  G
)  ->  F  e.  ( H  GrpAct  X ) )
Distinct variable groups:    x, y, G    x, X, y    x, Y, y    x,  .+ , y
Allowed substitution hints:    F( x, y)    H( x, y)

Proof of Theorem subgga
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subgga.3 . . . 4  |-  H  =  ( Gs  Y )
21subggrp 16078 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  H  e.  Grp )
3 subgga.1 . . . 4  |-  X  =  ( Base `  G
)
4 fvex 5866 . . . 4  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2527 . . 3  |-  X  e. 
_V
62, 5jctir 538 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( H  e.  Grp  /\  X  e. 
_V ) )
7 subgrcl 16080 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
87adantr 465 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  G  e.  Grp )
93subgss 16076 . . . . . . . . 9  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
109sselda 3489 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  Y )  ->  x  e.  X )
1110adantrr 716 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  x  e.  X )
12 simprr 757 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  y  e.  X )
13 subgga.2 . . . . . . . 8  |-  .+  =  ( +g  `  G )
143, 13grpcl 15937 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  y  e.  X )  ->  ( x  .+  y
)  e.  X )
158, 11, 12, 14syl3anc 1229 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  ( x  .+  y )  e.  X
)
1615ralrimivva 2864 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  A. x  e.  Y  A. y  e.  X  ( x  .+  y )  e.  X
)
17 subgga.4 . . . . . 6  |-  F  =  ( x  e.  Y ,  y  e.  X  |->  ( x  .+  y
) )
1817fmpt2 6852 . . . . 5  |-  ( A. x  e.  Y  A. y  e.  X  (
x  .+  y )  e.  X  <->  F : ( Y  X.  X ) --> X )
1916, 18sylib 196 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  F :
( Y  X.  X
) --> X )
201subgbas 16079 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  =  ( Base `  H )
)
2120xpeq1d 5012 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( Y  X.  X )  =  ( ( Base `  H
)  X.  X ) )
2221feq2d 5708 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( F : ( Y  X.  X ) --> X  <->  F :
( ( Base `  H
)  X.  X ) --> X ) )
2319, 22mpbid 210 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  F :
( ( Base `  H
)  X.  X ) --> X )
24 eqid 2443 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
2524subg0cl 16083 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  Y
)
26 oveq12 6290 . . . . . . . 8  |-  ( ( x  =  ( 0g
`  G )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( 0g `  G ) 
.+  u ) )
27 ovex 6309 . . . . . . . 8  |-  ( ( 0g `  G ) 
.+  u )  e. 
_V
2826, 17, 27ovmpt2a 6418 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  Y  /\  u  e.  X )  ->  ( ( 0g `  G ) F u )  =  ( ( 0g `  G ) 
.+  u ) )
2925, 28sylan 471 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
) F u )  =  ( ( 0g
`  G )  .+  u ) )
301, 24subg0 16081 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
3130oveq1d 6296 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( ( 0g `  G ) F u )  =  ( ( 0g `  H
) F u ) )
3231adantr 465 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
) F u )  =  ( ( 0g
`  H ) F u ) )
333, 13, 24grplid 15954 . . . . . . 7  |-  ( ( G  e.  Grp  /\  u  e.  X )  ->  ( ( 0g `  G )  .+  u
)  =  u )
347, 33sylan 471 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
)  .+  u )  =  u )
3529, 32, 343eqtr3d 2492 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  H
) F u )  =  u )
367ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  G  e.  Grp )
379ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  Y  C_  X
)
38 simprl 756 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  v  e.  Y )
3937, 38sseldd 3490 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  v  e.  X )
40 simprr 757 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  w  e.  Y )
4137, 40sseldd 3490 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  w  e.  X )
42 simplr 755 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  u  e.  X )
433, 13grpass 15938 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  ( v  e.  X  /\  w  e.  X  /\  u  e.  X
) )  ->  (
( v  .+  w
)  .+  u )  =  ( v  .+  ( w  .+  u ) ) )
4436, 39, 41, 42, 43syl13anc 1231 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v  .+  ( w 
.+  u ) ) )
453, 13grpcl 15937 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  u  e.  X )  ->  ( w  .+  u
)  e.  X )
4636, 41, 42, 45syl3anc 1229 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( w  .+  u )  e.  X
)
47 oveq12 6290 . . . . . . . . . . 11  |-  ( ( x  =  v  /\  y  =  ( w  .+  u ) )  -> 
( x  .+  y
)  =  ( v 
.+  ( w  .+  u ) ) )
48 ovex 6309 . . . . . . . . . . 11  |-  ( v 
.+  ( w  .+  u ) )  e. 
_V
4947, 17, 48ovmpt2a 6418 . . . . . . . . . 10  |-  ( ( v  e.  Y  /\  ( w  .+  u )  e.  X )  -> 
( v F ( w  .+  u ) )  =  ( v 
.+  ( w  .+  u ) ) )
5038, 46, 49syl2anc 661 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v F ( w  .+  u ) )  =  ( v  .+  (
w  .+  u )
) )
5144, 50eqtr4d 2487 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v F ( w 
.+  u ) ) )
5213subgcl 16085 . . . . . . . . . . 11  |-  ( ( Y  e.  (SubGrp `  G )  /\  v  e.  Y  /\  w  e.  Y )  ->  (
v  .+  w )  e.  Y )
53523expb 1198 . . . . . . . . . 10  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v  .+  w )  e.  Y
)
5453adantlr 714 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v  .+  w )  e.  Y
)
55 oveq12 6290 . . . . . . . . . 10  |-  ( ( x  =  ( v 
.+  w )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( v  .+  w ) 
.+  u ) )
56 ovex 6309 . . . . . . . . . 10  |-  ( ( v  .+  w ) 
.+  u )  e. 
_V
5755, 17, 56ovmpt2a 6418 . . . . . . . . 9  |-  ( ( ( v  .+  w
)  e.  Y  /\  u  e.  X )  ->  ( ( v  .+  w ) F u )  =  ( ( v  .+  w ) 
.+  u ) )
5854, 42, 57syl2anc 661 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w ) F u )  =  ( ( v  .+  w )  .+  u
) )
59 oveq12 6290 . . . . . . . . . . 11  |-  ( ( x  =  w  /\  y  =  u )  ->  ( x  .+  y
)  =  ( w 
.+  u ) )
60 ovex 6309 . . . . . . . . . . 11  |-  ( w 
.+  u )  e. 
_V
6159, 17, 60ovmpt2a 6418 . . . . . . . . . 10  |-  ( ( w  e.  Y  /\  u  e.  X )  ->  ( w F u )  =  ( w 
.+  u ) )
6240, 42, 61syl2anc 661 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( w F u )  =  ( w  .+  u
) )
6362oveq2d 6297 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v F ( w F u ) )  =  ( v F ( w  .+  u ) ) )
6451, 58, 633eqtr4d 2494 . . . . . . 7  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w ) F u )  =  ( v F ( w F u ) ) )
6564ralrimivva 2864 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  A. v  e.  Y  A. w  e.  Y  ( (
v  .+  w ) F u )  =  ( v F ( w F u ) ) )
661, 13ressplusg 14616 . . . . . . . . . . . 12  |-  ( Y  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  H ) )
6766oveqd 6298 . . . . . . . . . . 11  |-  ( Y  e.  (SubGrp `  G
)  ->  ( v  .+  w )  =  ( v ( +g  `  H
) w ) )
6867oveq1d 6296 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
v  .+  w ) F u )  =  ( ( v ( +g  `  H ) w ) F u ) )
6968eqeq1d 2445 . . . . . . . . 9  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
( v  .+  w
) F u )  =  ( v F ( w F u ) )  <->  ( (
v ( +g  `  H
) w ) F u )  =  ( v F ( w F u ) ) ) )
7020, 69raleqbidv 3054 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  ( A. w  e.  Y  (
( v  .+  w
) F u )  =  ( v F ( w F u ) )  <->  A. w  e.  ( Base `  H
) ( ( v ( +g  `  H
) w ) F u )  =  ( v F ( w F u ) ) ) )
7120, 70raleqbidv 3054 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( A. v  e.  Y  A. w  e.  Y  (
( v  .+  w
) F u )  =  ( v F ( w F u ) )  <->  A. v  e.  ( Base `  H
) A. w  e.  ( Base `  H
) ( ( v ( +g  `  H
) w ) F u )  =  ( v F ( w F u ) ) ) )
7271biimpa 484 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  A. v  e.  Y  A. w  e.  Y  (
( v  .+  w
) F u )  =  ( v F ( w F u ) ) )  ->  A. v  e.  ( Base `  H ) A. w  e.  ( Base `  H ) ( ( v ( +g  `  H
) w ) F u )  =  ( v F ( w F u ) ) )
7365, 72syldan 470 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  A. v  e.  ( Base `  H
) A. w  e.  ( Base `  H
) ( ( v ( +g  `  H
) w ) F u )  =  ( v F ( w F u ) ) )
7435, 73jca 532 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( ( 0g `  H ) F u )  =  u  /\  A. v  e.  ( Base `  H ) A. w  e.  ( Base `  H
) ( ( v ( +g  `  H
) w ) F u )  =  ( v F ( w F u ) ) ) )
7574ralrimiva 2857 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  A. u  e.  X  ( (
( 0g `  H
) F u )  =  u  /\  A. v  e.  ( Base `  H ) A. w  e.  ( Base `  H
) ( ( v ( +g  `  H
) w ) F u )  =  ( v F ( w F u ) ) ) )
7623, 75jca 532 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( F : ( ( Base `  H )  X.  X
) --> X  /\  A. u  e.  X  (
( ( 0g `  H ) F u )  =  u  /\  A. v  e.  ( Base `  H ) A. w  e.  ( Base `  H
) ( ( v ( +g  `  H
) w ) F u )  =  ( v F ( w F u ) ) ) ) )
77 eqid 2443 . . 3  |-  ( Base `  H )  =  (
Base `  H )
78 eqid 2443 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
79 eqid 2443 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
8077, 78, 79isga 16203 . 2  |-  ( F  e.  ( H  GrpAct  X )  <->  ( ( H  e.  Grp  /\  X  e.  _V )  /\  ( F : ( ( Base `  H )  X.  X
) --> X  /\  A. u  e.  X  (
( ( 0g `  H ) F u )  =  u  /\  A. v  e.  ( Base `  H ) A. w  e.  ( Base `  H
) ( ( v ( +g  `  H
) w ) F u )  =  ( v F ( w F u ) ) ) ) ) )
816, 76, 80sylanbrc 664 1  |-  ( Y  e.  (SubGrp `  G
)  ->  F  e.  ( H  GrpAct  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    C_ wss 3461    X. cxp 4987   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Basecbs 14509   ↾s cress 14510   +g cplusg 14574   0gc0g 14714   Grpcgrp 15927  SubGrpcsubg 16069    GrpAct cga 16201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-subg 16072  df-ga 16202
This theorem is referenced by:  gaid2  16215
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