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Theorem subgga 16954
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 16820 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  H  e.  Grp )
3 subgga.1 . . . 4  |-  X  =  ( Base `  G
)
4 fvex 5875 . . . 4  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2525 . . 3  |-  X  e. 
_V
62, 5jctir 541 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( H  e.  Grp  /\  X  e. 
_V ) )
7 subgrcl 16822 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
87adantr 467 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  G  e.  Grp )
93subgss 16818 . . . . . . . . 9  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
109sselda 3432 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  Y )  ->  x  e.  X )
1110adantrr 723 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  x  e.  X )
12 simprr 766 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  y  e.  X )
13 subgga.2 . . . . . . . 8  |-  .+  =  ( +g  `  G )
143, 13grpcl 16679 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  y  e.  X )  ->  ( x  .+  y
)  e.  X )
158, 11, 12, 14syl3anc 1268 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  ( x  .+  y )  e.  X
)
1615ralrimivva 2809 . . . . 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 6860 . . . . 5  |-  ( A. x  e.  Y  A. y  e.  X  (
x  .+  y )  e.  X  <->  F : ( Y  X.  X ) --> X )
1916, 18sylib 200 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  F :
( Y  X.  X
) --> X )
201subgbas 16821 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  =  ( Base `  H )
)
2120xpeq1d 4857 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( Y  X.  X )  =  ( ( Base `  H
)  X.  X ) )
2221feq2d 5715 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( F : ( Y  X.  X ) --> X  <->  F :
( ( Base `  H
)  X.  X ) --> X ) )
2319, 22mpbid 214 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  F :
( ( Base `  H
)  X.  X ) --> X )
24 eqid 2451 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
2524subg0cl 16825 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  Y
)
26 oveq12 6299 . . . . . . . 8  |-  ( ( x  =  ( 0g
`  G )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( 0g `  G ) 
.+  u ) )
27 ovex 6318 . . . . . . . 8  |-  ( ( 0g `  G ) 
.+  u )  e. 
_V
2826, 17, 27ovmpt2a 6427 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  Y  /\  u  e.  X )  ->  ( ( 0g `  G ) F u )  =  ( ( 0g `  G ) 
.+  u ) )
2925, 28sylan 474 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
) F u )  =  ( ( 0g
`  G )  .+  u ) )
301, 24subg0 16823 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
3130oveq1d 6305 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( ( 0g `  G ) F u )  =  ( ( 0g `  H
) F u ) )
3231adantr 467 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
) F u )  =  ( ( 0g
`  H ) F u ) )
333, 13, 24grplid 16696 . . . . . . 7  |-  ( ( G  e.  Grp  /\  u  e.  X )  ->  ( ( 0g `  G )  .+  u
)  =  u )
347, 33sylan 474 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
)  .+  u )  =  u )
3529, 32, 343eqtr3d 2493 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  H
) F u )  =  u )
367ad2antrr 732 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  G  e.  Grp )
379ad2antrr 732 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  Y  C_  X
)
38 simprl 764 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  v  e.  Y )
3937, 38sseldd 3433 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  v  e.  X )
40 simprr 766 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  w  e.  Y )
4137, 40sseldd 3433 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  w  e.  X )
42 simplr 762 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  u  e.  X )
433, 13grpass 16680 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  ( v  e.  X  /\  w  e.  X  /\  u  e.  X
) )  ->  (
( v  .+  w
)  .+  u )  =  ( v  .+  ( w  .+  u ) ) )
4436, 39, 41, 42, 43syl13anc 1270 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v  .+  ( w 
.+  u ) ) )
453, 13grpcl 16679 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  u  e.  X )  ->  ( w  .+  u
)  e.  X )
4636, 41, 42, 45syl3anc 1268 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( w  .+  u )  e.  X
)
47 oveq12 6299 . . . . . . . . . . 11  |-  ( ( x  =  v  /\  y  =  ( w  .+  u ) )  -> 
( x  .+  y
)  =  ( v 
.+  ( w  .+  u ) ) )
48 ovex 6318 . . . . . . . . . . 11  |-  ( v 
.+  ( w  .+  u ) )  e. 
_V
4947, 17, 48ovmpt2a 6427 . . . . . . . . . 10  |-  ( ( v  e.  Y  /\  ( w  .+  u )  e.  X )  -> 
( v F ( w  .+  u ) )  =  ( v 
.+  ( w  .+  u ) ) )
5038, 46, 49syl2anc 667 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v F ( w  .+  u ) )  =  ( v  .+  (
w  .+  u )
) )
5144, 50eqtr4d 2488 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v F ( w 
.+  u ) ) )
5213subgcl 16827 . . . . . . . . . . 11  |-  ( ( Y  e.  (SubGrp `  G )  /\  v  e.  Y  /\  w  e.  Y )  ->  (
v  .+  w )  e.  Y )
53523expb 1209 . . . . . . . . . 10  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v  .+  w )  e.  Y
)
5453adantlr 721 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v  .+  w )  e.  Y
)
55 oveq12 6299 . . . . . . . . . 10  |-  ( ( x  =  ( v 
.+  w )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( v  .+  w ) 
.+  u ) )
56 ovex 6318 . . . . . . . . . 10  |-  ( ( v  .+  w ) 
.+  u )  e. 
_V
5755, 17, 56ovmpt2a 6427 . . . . . . . . 9  |-  ( ( ( v  .+  w
)  e.  Y  /\  u  e.  X )  ->  ( ( v  .+  w ) F u )  =  ( ( v  .+  w ) 
.+  u ) )
5854, 42, 57syl2anc 667 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w ) F u )  =  ( ( v  .+  w )  .+  u
) )
59 oveq12 6299 . . . . . . . . . . 11  |-  ( ( x  =  w  /\  y  =  u )  ->  ( x  .+  y
)  =  ( w 
.+  u ) )
60 ovex 6318 . . . . . . . . . . 11  |-  ( w 
.+  u )  e. 
_V
6159, 17, 60ovmpt2a 6427 . . . . . . . . . 10  |-  ( ( w  e.  Y  /\  u  e.  X )  ->  ( w F u )  =  ( w 
.+  u ) )
6240, 42, 61syl2anc 667 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( w F u )  =  ( w  .+  u
) )
6362oveq2d 6306 . . . . . . . 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 2495 . . . . . . 7  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w ) F u )  =  ( v F ( w F u ) ) )
6564ralrimivva 2809 . . . . . 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 15239 . . . . . . . . . . . 12  |-  ( Y  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  H ) )
6766oveqd 6307 . . . . . . . . . . 11  |-  ( Y  e.  (SubGrp `  G
)  ->  ( v  .+  w )  =  ( v ( +g  `  H
) w ) )
6867oveq1d 6305 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
v  .+  w ) F u )  =  ( ( v ( +g  `  H ) w ) F u ) )
6968eqeq1d 2453 . . . . . . . . 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 3001 . . . . . . . 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 3001 . . . . . . 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 487 . . . . . 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 473 . . . . 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 535 . . . 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 2802 . . 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 535 . 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 2451 . . 3  |-  ( Base `  H )  =  (
Base `  H )
78 eqid 2451 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
79 eqid 2451 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
8077, 78, 79isga 16945 . 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 670 1  |-  ( Y  e.  (SubGrp `  G
)  ->  F  e.  ( H  GrpAct  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    C_ wss 3404    X. cxp 4832   -->wf 5578   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   Basecbs 15121   ↾s cress 15122   +g cplusg 15190   0gc0g 15338   Grpcgrp 16669  SubGrpcsubg 16811    GrpAct cga 16943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-subg 16814  df-ga 16944
This theorem is referenced by:  gaid2  16957
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