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Theorem subgga 16133
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 15999 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  H  e.  Grp )
3 subgga.1 . . . 4  |-  X  =  ( Base `  G
)
4 fvex 5874 . . . 4  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2551 . . 3  |-  X  e. 
_V
62, 5jctir 538 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( H  e.  Grp  /\  X  e. 
_V ) )
7 subgrcl 16001 . . . . . . . 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 15997 . . . . . . . . 9  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
109sselda 3504 . . . . . . . 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 756 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  y  e.  X )
13 subgga.2 . . . . . . . 8  |-  .+  =  ( +g  `  G )
143, 13grpcl 15864 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  y  e.  X )  ->  ( x  .+  y
)  e.  X )
158, 11, 12, 14syl3anc 1228 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  ( x  .+  y )  e.  X
)
1615ralrimivva 2885 . . . . 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 6848 . . . . 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 16000 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  =  ( Base `  H )
)
2120xpeq1d 5022 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( Y  X.  X )  =  ( ( Base `  H
)  X.  X ) )
2221feq2d 5716 . . . 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 2467 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
2524subg0cl 16004 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  Y
)
26 oveq12 6291 . . . . . . . 8  |-  ( ( x  =  ( 0g
`  G )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( 0g `  G ) 
.+  u ) )
27 ovex 6307 . . . . . . . 8  |-  ( ( 0g `  G ) 
.+  u )  e. 
_V
2826, 17, 27ovmpt2a 6415 . . . . . . 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 16002 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
3130oveq1d 6297 . . . . . . 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 15881 . . . . . . 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 2516 . . . . 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 755 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  v  e.  Y )
3937, 38sseldd 3505 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  v  e.  X )
40 simprr 756 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  w  e.  Y )
4137, 40sseldd 3505 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  w  e.  X )
42 simplr 754 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  u  e.  X )
433, 13grpass 15865 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  ( v  e.  X  /\  w  e.  X  /\  u  e.  X
) )  ->  (
( v  .+  w
)  .+  u )  =  ( v  .+  ( w  .+  u ) ) )
4436, 39, 41, 42, 43syl13anc 1230 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v  .+  ( w 
.+  u ) ) )
453, 13grpcl 15864 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  u  e.  X )  ->  ( w  .+  u
)  e.  X )
4636, 41, 42, 45syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( w  .+  u )  e.  X
)
47 oveq12 6291 . . . . . . . . . . 11  |-  ( ( x  =  v  /\  y  =  ( w  .+  u ) )  -> 
( x  .+  y
)  =  ( v 
.+  ( w  .+  u ) ) )
48 ovex 6307 . . . . . . . . . . 11  |-  ( v 
.+  ( w  .+  u ) )  e. 
_V
4947, 17, 48ovmpt2a 6415 . . . . . . . . . 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 2511 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v F ( w 
.+  u ) ) )
5213subgcl 16006 . . . . . . . . . . 11  |-  ( ( Y  e.  (SubGrp `  G )  /\  v  e.  Y  /\  w  e.  Y )  ->  (
v  .+  w )  e.  Y )
53523expb 1197 . . . . . . . . . 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 6291 . . . . . . . . . 10  |-  ( ( x  =  ( v 
.+  w )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( v  .+  w ) 
.+  u ) )
56 ovex 6307 . . . . . . . . . 10  |-  ( ( v  .+  w ) 
.+  u )  e. 
_V
5755, 17, 56ovmpt2a 6415 . . . . . . . . 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 6291 . . . . . . . . . . 11  |-  ( ( x  =  w  /\  y  =  u )  ->  ( x  .+  y
)  =  ( w 
.+  u ) )
60 ovex 6307 . . . . . . . . . . 11  |-  ( w 
.+  u )  e. 
_V
6159, 17, 60ovmpt2a 6415 . . . . . . . . . 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 6298 . . . . . . . 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 2518 . . . . . . 7  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w ) F u )  =  ( v F ( w F u ) ) )
6564ralrimivva 2885 . . . . . 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 14593 . . . . . . . . . . . 12  |-  ( Y  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  H ) )
6766oveqd 6299 . . . . . . . . . . 11  |-  ( Y  e.  (SubGrp `  G
)  ->  ( v  .+  w )  =  ( v ( +g  `  H
) w ) )
6867oveq1d 6297 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
v  .+  w ) F u )  =  ( ( v ( +g  `  H ) w ) F u ) )
6968eqeq1d 2469 . . . . . . . . 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 3072 . . . . . . . 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 3072 . . . . . . 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 2878 . . 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 2467 . . 3  |-  ( Base `  H )  =  (
Base `  H )
78 eqid 2467 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
79 eqid 2467 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
8077, 78, 79isga 16124 . 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 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476    X. cxp 4997   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   Basecbs 14486   ↾s cress 14487   +g cplusg 14551   0gc0g 14691   Grpcgrp 15723  SubGrpcsubg 15990    GrpAct cga 16122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-mnd 15728  df-grp 15858  df-subg 15993  df-ga 16123
This theorem is referenced by:  gaid2  16136
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