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Theorem eqgval 14668
Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
eqgval.x  |-  X  =  ( Base `  G
)
eqgval.n  |-  N  =  ( inv g `  G )
eqgval.p  |-  .+  =  ( +g  `  G )
eqgval.r  |-  R  =  ( G ~QG  S )
Assertion
Ref Expression
eqgval  |-  ( ( G  e.  V  /\  S  C_  X )  -> 
( A R B  <-> 
( A  e.  X  /\  B  e.  X  /\  ( ( N `  A )  .+  B
)  e.  S ) ) )

Proof of Theorem eqgval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqgval.x . . . 4  |-  X  =  ( Base `  G
)
2 eqgval.n . . . 4  |-  N  =  ( inv g `  G )
3 eqgval.p . . . 4  |-  .+  =  ( +g  `  G )
4 eqgval.r . . . 4  |-  R  =  ( G ~QG  S )
51, 2, 3, 4eqgfval 14667 . . 3  |-  ( ( G  e.  V  /\  S  C_  X )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } )
65breqd 4036 . 2  |-  ( ( G  e.  V  /\  S  C_  X )  -> 
( A R B  <-> 
A { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } B ) )
7 relopab 4814 . . . . . 6  |-  Rel  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `
 x )  .+  y )  e.  S
) }
87brrelexi 4731 . . . . 5  |-  ( A { <. x ,  y
>.  |  ( {
x ,  y } 
C_  X  /\  (
( N `  x
)  .+  y )  e.  S ) } B  ->  A  e.  _V )
97brrelex2i 4732 . . . . 5  |-  ( A { <. x ,  y
>.  |  ( {
x ,  y } 
C_  X  /\  (
( N `  x
)  .+  y )  e.  S ) } B  ->  B  e.  _V )
108, 9jca 518 . . . 4  |-  ( A { <. x ,  y
>.  |  ( {
x ,  y } 
C_  X  /\  (
( N `  x
)  .+  y )  e.  S ) } B  ->  ( A  e.  _V  /\  B  e.  _V )
)
1110adantl 452 . . 3  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  A { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
12 simpr1 961 . . . . 5  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  A  e.  X )
13 elex 2798 . . . . 5  |-  ( A  e.  X  ->  A  e.  _V )
1412, 13syl 15 . . . 4  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  A  e.  _V )
15 simpr2 962 . . . . 5  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  B  e.  X )
16 elex 2798 . . . . 5  |-  ( B  e.  X  ->  B  e.  _V )
1715, 16syl 15 . . . 4  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  B  e.  _V )
1814, 17jca 518 . . 3  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
19 vex 2793 . . . . . . . 8  |-  x  e. 
_V
20 vex 2793 . . . . . . . 8  |-  y  e. 
_V
2119, 20prss 3771 . . . . . . 7  |-  ( ( x  e.  X  /\  y  e.  X )  <->  { x ,  y } 
C_  X )
22 eleq1 2345 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  X  <->  A  e.  X ) )
23 eleq1 2345 . . . . . . . 8  |-  ( y  =  B  ->  (
y  e.  X  <->  B  e.  X ) )
2422, 23bi2anan9 843 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  X  /\  y  e.  X )  <->  ( A  e.  X  /\  B  e.  X ) ) )
2521, 24syl5bbr 250 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( { x ,  y }  C_  X  <->  ( A  e.  X  /\  B  e.  X )
) )
26 fveq2 5527 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
27 id 19 . . . . . . . 8  |-  ( y  =  B  ->  y  =  B )
2826, 27oveqan12d 5879 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( N `  x )  .+  y
)  =  ( ( N `  A ) 
.+  B ) )
2928eleq1d 2351 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( N `
 x )  .+  y )  e.  S  <->  ( ( N `  A
)  .+  B )  e.  S ) )
3025, 29anbi12d 691 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( { x ,  y }  C_  X  /\  ( ( N `
 x )  .+  y )  e.  S
)  <->  ( ( A  e.  X  /\  B  e.  X )  /\  (
( N `  A
)  .+  B )  e.  S ) ) )
31 df-3an 936 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A )  .+  B
)  e.  S )  <-> 
( ( A  e.  X  /\  B  e.  X )  /\  (
( N `  A
)  .+  B )  e.  S ) )
3230, 31syl6bbr 254 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( { x ,  y }  C_  X  /\  ( ( N `
 x )  .+  y )  e.  S
)  <->  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) ) )
33 eqid 2285 . . . 4  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }
3432, 33brabga 4281 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } B  <->  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) ) )
3511, 18, 34pm5.21nd 868 . 2  |-  ( ( G  e.  V  /\  S  C_  X )  -> 
( A { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } B  <->  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) ) )
366, 35bitrd 244 1  |-  ( ( G  e.  V  /\  S  C_  X )  -> 
( A R B  <-> 
( A  e.  X  /\  B  e.  X  /\  ( ( N `  A )  .+  B
)  e.  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   _Vcvv 2790    C_ wss 3154   {cpr 3643   class class class wbr 4025   {copab 4078   ` cfv 5257  (class class class)co 5860   Basecbs 13150   +g cplusg 13210   inv gcminusg 14365   ~QG cqg 14619
This theorem is referenced by:  eqger  14669  eqglact  14670  eqgid  14671  eqgcpbl  14673  gastacos  14766  orbstafun  14767  sylow2blem1  14933  sylow2blem3  14935  eqgabl  15133  tgpconcompeqg  17796  tgpconcomp  17797  divstgpopn  17804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-iota 5221  df-fun 5259  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-eqg 14622
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