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Theorem subgga 15811
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 15677 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  H  e.  Grp )
3 subgga.1 . . . 4  |-  X  =  ( Base `  G
)
4 fvex 5698 . . . 4  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2511 . . 3  |-  X  e. 
_V
62, 5jctir 535 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( H  e.  Grp  /\  X  e. 
_V ) )
7 subgrcl 15679 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
87adantr 462 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  G  e.  Grp )
93subgss 15675 . . . . . . . . 9  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
109sselda 3353 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  Y )  ->  x  e.  X )
1110adantrr 711 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  x  e.  X )
12 simprr 751 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  y  e.  X )
13 subgga.2 . . . . . . . 8  |-  .+  =  ( +g  `  G )
143, 13grpcl 15544 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  y  e.  X )  ->  ( x  .+  y
)  e.  X )
158, 11, 12, 14syl3anc 1213 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  ( x  .+  y )  e.  X
)
1615ralrimivva 2806 . . . . 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 6640 . . . . 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 15678 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  =  ( Base `  H )
)
2120xpeq1d 4859 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( Y  X.  X )  =  ( ( Base `  H
)  X.  X ) )
2221feq2d 5544 . . . 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 2441 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
2524subg0cl 15682 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  Y
)
26 oveq12 6099 . . . . . . . 8  |-  ( ( x  =  ( 0g
`  G )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( 0g `  G ) 
.+  u ) )
27 ovex 6115 . . . . . . . 8  |-  ( ( 0g `  G ) 
.+  u )  e. 
_V
2826, 17, 27ovmpt2a 6220 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  Y  /\  u  e.  X )  ->  ( ( 0g `  G ) F u )  =  ( ( 0g `  G ) 
.+  u ) )
2925, 28sylan 468 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
) F u )  =  ( ( 0g
`  G )  .+  u ) )
301, 24subg0 15680 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
3130oveq1d 6105 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( ( 0g `  G ) F u )  =  ( ( 0g `  H
) F u ) )
3231adantr 462 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
) F u )  =  ( ( 0g
`  H ) F u ) )
333, 13, 24grplid 15561 . . . . . . 7  |-  ( ( G  e.  Grp  /\  u  e.  X )  ->  ( ( 0g `  G )  .+  u
)  =  u )
347, 33sylan 468 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
)  .+  u )  =  u )
3529, 32, 343eqtr3d 2481 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  H
) F u )  =  u )
367ad2antrr 720 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  G  e.  Grp )
379ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  Y  C_  X
)
38 simprl 750 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  v  e.  Y )
3937, 38sseldd 3354 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  v  e.  X )
40 simprr 751 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  w  e.  Y )
4137, 40sseldd 3354 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  w  e.  X )
42 simplr 749 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  u  e.  X )
433, 13grpass 15545 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  ( v  e.  X  /\  w  e.  X  /\  u  e.  X
) )  ->  (
( v  .+  w
)  .+  u )  =  ( v  .+  ( w  .+  u ) ) )
4436, 39, 41, 42, 43syl13anc 1215 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v  .+  ( w 
.+  u ) ) )
453, 13grpcl 15544 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  u  e.  X )  ->  ( w  .+  u
)  e.  X )
4636, 41, 42, 45syl3anc 1213 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( w  .+  u )  e.  X
)
47 oveq12 6099 . . . . . . . . . . 11  |-  ( ( x  =  v  /\  y  =  ( w  .+  u ) )  -> 
( x  .+  y
)  =  ( v 
.+  ( w  .+  u ) ) )
48 ovex 6115 . . . . . . . . . . 11  |-  ( v 
.+  ( w  .+  u ) )  e. 
_V
4947, 17, 48ovmpt2a 6220 . . . . . . . . . 10  |-  ( ( v  e.  Y  /\  ( w  .+  u )  e.  X )  -> 
( v F ( w  .+  u ) )  =  ( v 
.+  ( w  .+  u ) ) )
5038, 46, 49syl2anc 656 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v F ( w  .+  u ) )  =  ( v  .+  (
w  .+  u )
) )
5144, 50eqtr4d 2476 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v F ( w 
.+  u ) ) )
5213subgcl 15684 . . . . . . . . . . 11  |-  ( ( Y  e.  (SubGrp `  G )  /\  v  e.  Y  /\  w  e.  Y )  ->  (
v  .+  w )  e.  Y )
53523expb 1183 . . . . . . . . . 10  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v  .+  w )  e.  Y
)
5453adantlr 709 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v  .+  w )  e.  Y
)
55 oveq12 6099 . . . . . . . . . 10  |-  ( ( x  =  ( v 
.+  w )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( v  .+  w ) 
.+  u ) )
56 ovex 6115 . . . . . . . . . 10  |-  ( ( v  .+  w ) 
.+  u )  e. 
_V
5755, 17, 56ovmpt2a 6220 . . . . . . . . 9  |-  ( ( ( v  .+  w
)  e.  Y  /\  u  e.  X )  ->  ( ( v  .+  w ) F u )  =  ( ( v  .+  w ) 
.+  u ) )
5854, 42, 57syl2anc 656 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w ) F u )  =  ( ( v  .+  w )  .+  u
) )
59 oveq12 6099 . . . . . . . . . . 11  |-  ( ( x  =  w  /\  y  =  u )  ->  ( x  .+  y
)  =  ( w 
.+  u ) )
60 ovex 6115 . . . . . . . . . . 11  |-  ( w 
.+  u )  e. 
_V
6159, 17, 60ovmpt2a 6220 . . . . . . . . . 10  |-  ( ( w  e.  Y  /\  u  e.  X )  ->  ( w F u )  =  ( w 
.+  u ) )
6240, 42, 61syl2anc 656 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( w F u )  =  ( w  .+  u
) )
6362oveq2d 6106 . . . . . . . 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 2483 . . . . . . 7  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w ) F u )  =  ( v F ( w F u ) ) )
6564ralrimivva 2806 . . . . . 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 14276 . . . . . . . . . . . 12  |-  ( Y  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  H ) )
6766oveqd 6107 . . . . . . . . . . 11  |-  ( Y  e.  (SubGrp `  G
)  ->  ( v  .+  w )  =  ( v ( +g  `  H
) w ) )
6867oveq1d 6105 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
v  .+  w ) F u )  =  ( ( v ( +g  `  H ) w ) F u ) )
6968eqeq1d 2449 . . . . . . . . 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 2929 . . . . . . . 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 2929 . . . . . . 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 481 . . . . . 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 467 . . . . 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 529 . . . 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 2797 . . 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 529 . 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 2441 . . 3  |-  ( Base `  H )  =  (
Base `  H )
78 eqid 2441 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
79 eqid 2441 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
8077, 78, 79isga 15802 . 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 659 1  |-  ( Y  e.  (SubGrp `  G
)  ->  F  e.  ( H  GrpAct  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    C_ wss 3325    X. cxp 4834   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   Basecbs 14170   ↾s cress 14171   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406  SubGrpcsubg 15668    GrpAct cga 15800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mnd 15411  df-grp 15538  df-subg 15671  df-ga 15801
This theorem is referenced by:  gaid2  15814
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