MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgga Structured version   Unicode version

Theorem subgga 16537
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 16403 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  H  e.  Grp )
3 subgga.1 . . . 4  |-  X  =  ( Base `  G
)
4 fvex 5858 . . . 4  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2538 . . 3  |-  X  e. 
_V
62, 5jctir 536 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( H  e.  Grp  /\  X  e. 
_V ) )
7 subgrcl 16405 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
87adantr 463 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  G  e.  Grp )
93subgss 16401 . . . . . . . . 9  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
109sselda 3489 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  Y )  ->  x  e.  X )
1110adantrr 714 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  x  e.  X )
12 simprr 755 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  y  e.  X )
13 subgga.2 . . . . . . . 8  |-  .+  =  ( +g  `  G )
143, 13grpcl 16262 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  y  e.  X )  ->  ( x  .+  y
)  e.  X )
158, 11, 12, 14syl3anc 1226 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
x  e.  Y  /\  y  e.  X )
)  ->  ( x  .+  y )  e.  X
)
1615ralrimivva 2875 . . . . 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 6840 . . . . 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 16404 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  =  ( Base `  H )
)
2120xpeq1d 5011 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( Y  X.  X )  =  ( ( Base `  H
)  X.  X ) )
2221feq2d 5700 . . . 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 2454 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
2524subg0cl 16408 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  Y
)
26 oveq12 6279 . . . . . . . 8  |-  ( ( x  =  ( 0g
`  G )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( 0g `  G ) 
.+  u ) )
27 ovex 6298 . . . . . . . 8  |-  ( ( 0g `  G ) 
.+  u )  e. 
_V
2826, 17, 27ovmpt2a 6406 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  Y  /\  u  e.  X )  ->  ( ( 0g `  G ) F u )  =  ( ( 0g `  G ) 
.+  u ) )
2925, 28sylan 469 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
) F u )  =  ( ( 0g
`  G )  .+  u ) )
301, 24subg0 16406 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
3130oveq1d 6285 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  ( ( 0g `  G ) F u )  =  ( ( 0g `  H
) F u ) )
3231adantr 463 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
) F u )  =  ( ( 0g
`  H ) F u ) )
333, 13, 24grplid 16279 . . . . . . 7  |-  ( ( G  e.  Grp  /\  u  e.  X )  ->  ( ( 0g `  G )  .+  u
)  =  u )
347, 33sylan 469 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  G
)  .+  u )  =  u )
3529, 32, 343eqtr3d 2503 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  ->  (
( 0g `  H
) F u )  =  u )
367ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  G  e.  Grp )
379ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  Y  C_  X
)
38 simprl 754 . . . . . . . . . . 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 755 . . . . . . . . . . 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 753 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  u  e.  X )
433, 13grpass 16263 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  ( v  e.  X  /\  w  e.  X  /\  u  e.  X
) )  ->  (
( v  .+  w
)  .+  u )  =  ( v  .+  ( w  .+  u ) ) )
4436, 39, 41, 42, 43syl13anc 1228 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v  .+  ( w 
.+  u ) ) )
453, 13grpcl 16262 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  u  e.  X )  ->  ( w  .+  u
)  e.  X )
4636, 41, 42, 45syl3anc 1226 . . . . . . . . . 10  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( w  .+  u )  e.  X
)
47 oveq12 6279 . . . . . . . . . . 11  |-  ( ( x  =  v  /\  y  =  ( w  .+  u ) )  -> 
( x  .+  y
)  =  ( v 
.+  ( w  .+  u ) ) )
48 ovex 6298 . . . . . . . . . . 11  |-  ( v 
.+  ( w  .+  u ) )  e. 
_V
4947, 17, 48ovmpt2a 6406 . . . . . . . . . 10  |-  ( ( v  e.  Y  /\  ( w  .+  u )  e.  X )  -> 
( v F ( w  .+  u ) )  =  ( v 
.+  ( w  .+  u ) ) )
5038, 46, 49syl2anc 659 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v F ( w  .+  u ) )  =  ( v  .+  (
w  .+  u )
) )
5144, 50eqtr4d 2498 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w )  .+  u )  =  ( v F ( w 
.+  u ) ) )
5213subgcl 16410 . . . . . . . . . . 11  |-  ( ( Y  e.  (SubGrp `  G )  /\  v  e.  Y  /\  w  e.  Y )  ->  (
v  .+  w )  e.  Y )
53523expb 1195 . . . . . . . . . 10  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v  .+  w )  e.  Y
)
5453adantlr 712 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( v  .+  w )  e.  Y
)
55 oveq12 6279 . . . . . . . . . 10  |-  ( ( x  =  ( v 
.+  w )  /\  y  =  u )  ->  ( x  .+  y
)  =  ( ( v  .+  w ) 
.+  u ) )
56 ovex 6298 . . . . . . . . . 10  |-  ( ( v  .+  w ) 
.+  u )  e. 
_V
5755, 17, 56ovmpt2a 6406 . . . . . . . . 9  |-  ( ( ( v  .+  w
)  e.  Y  /\  u  e.  X )  ->  ( ( v  .+  w ) F u )  =  ( ( v  .+  w ) 
.+  u ) )
5854, 42, 57syl2anc 659 . . . . . . . 8  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w ) F u )  =  ( ( v  .+  w )  .+  u
) )
59 oveq12 6279 . . . . . . . . . . 11  |-  ( ( x  =  w  /\  y  =  u )  ->  ( x  .+  y
)  =  ( w 
.+  u ) )
60 ovex 6298 . . . . . . . . . . 11  |-  ( w 
.+  u )  e. 
_V
6159, 17, 60ovmpt2a 6406 . . . . . . . . . 10  |-  ( ( w  e.  Y  /\  u  e.  X )  ->  ( w F u )  =  ( w 
.+  u ) )
6240, 42, 61syl2anc 659 . . . . . . . . 9  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( w F u )  =  ( w  .+  u
) )
6362oveq2d 6286 . . . . . . . 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 2505 . . . . . . 7  |-  ( ( ( Y  e.  (SubGrp `  G )  /\  u  e.  X )  /\  (
v  e.  Y  /\  w  e.  Y )
)  ->  ( (
v  .+  w ) F u )  =  ( v F ( w F u ) ) )
6564ralrimivva 2875 . . . . . 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 14830 . . . . . . . . . . . 12  |-  ( Y  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  H ) )
6766oveqd 6287 . . . . . . . . . . 11  |-  ( Y  e.  (SubGrp `  G
)  ->  ( v  .+  w )  =  ( v ( +g  `  H
) w ) )
6867oveq1d 6285 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
v  .+  w ) F u )  =  ( ( v ( +g  `  H ) w ) F u ) )
6968eqeq1d 2456 . . . . . . . . 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 3065 . . . . . . . 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 3065 . . . . . . 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 482 . . . . . 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 468 . . . . 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 530 . . . 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 2868 . . 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 530 . 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 2454 . . 3  |-  ( Base `  H )  =  (
Base `  H )
78 eqid 2454 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
79 eqid 2454 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
8077, 78, 79isga 16528 . 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 662 1  |-  ( Y  e.  (SubGrp `  G
)  ->  F  e.  ( H  GrpAct  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    C_ wss 3461    X. cxp 4986   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Basecbs 14716   ↾s cress 14717   +g cplusg 14784   0gc0g 14929   Grpcgrp 16252  SubGrpcsubg 16394    GrpAct cga 16526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-subg 16397  df-ga 16527
This theorem is referenced by:  gaid2  16540
  Copyright terms: Public domain W3C validator