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Theorem eqgfval 14665
Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-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
eqgfval  |-  ( ( G  e.  V  /\  S  C_  X )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } )
Distinct variable groups:    x, y, G    x, N, y    x, S, y    x,  .+ , y    x, X, y
Allowed substitution hints:    R( x, y)    V( x, y)

Proof of Theorem eqgfval
Dummy variables  g 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 eqgval.x . . . 4  |-  X  =  ( Base `  G
)
3 fvex 5539 . . . 4  |-  ( Base `  G )  e.  _V
42, 3eqeltri 2353 . . 3  |-  X  e. 
_V
54ssex 4158 . 2  |-  ( S 
C_  X  ->  S  e.  _V )
6 eqgval.r . . 3  |-  R  =  ( G ~QG  S )
7 simpl 443 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  g  =  G )
87fveq2d 5529 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( Base `  g
)  =  ( Base `  G ) )
98, 2syl6eqr 2333 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  ( Base `  g
)  =  X )
109sseq2d 3206 . . . . . 6  |-  ( ( g  =  G  /\  s  =  S )  ->  ( { x ,  y }  C_  ( Base `  g )  <->  { x ,  y }  C_  X ) )
117fveq2d 5529 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  ( +g  `  g
)  =  ( +g  `  G ) )
12 eqgval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
1311, 12syl6eqr 2333 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( +g  `  g
)  =  .+  )
147fveq2d 5529 . . . . . . . . . 10  |-  ( ( g  =  G  /\  s  =  S )  ->  ( inv g `  g )  =  ( inv g `  G
) )
15 eqgval.n . . . . . . . . . 10  |-  N  =  ( inv g `  G )
1614, 15syl6eqr 2333 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  ( inv g `  g )  =  N )
1716fveq1d 5527 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( inv g `  g ) `  x
)  =  ( N `
 x ) )
18 eqidd 2284 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  y  =  y )
1913, 17, 18oveq123d 5879 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( ( inv g `  g ) `
 x ) ( +g  `  g ) y )  =  ( ( N `  x
)  .+  y )
)
20 simpr 447 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  s  =  S )
2119, 20eleq12d 2351 . . . . . 6  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( ( ( inv g `  g
) `  x )
( +g  `  g ) y )  e.  s  <-> 
( ( N `  x )  .+  y
)  e.  S ) )
2210, 21anbi12d 691 . . . . 5  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( { x ,  y }  C_  ( Base `  g )  /\  ( ( ( inv g `  g ) `
 x ) ( +g  `  g ) y )  e.  s )  <->  ( { x ,  y }  C_  X  /\  ( ( N `
 x )  .+  y )  e.  S
) ) )
2322opabbidv 4082 . . . 4  |-  ( ( g  =  G  /\  s  =  S )  ->  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( Base `  g
)  /\  ( (
( inv g `  g ) `  x
) ( +g  `  g
) y )  e.  s ) }  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  X  /\  (
( N `  x
)  .+  y )  e.  S ) } )
24 df-eqg 14620 . . . 4  |- ~QG  =  ( g  e.  _V ,  s  e. 
_V  |->  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  g )  /\  (
( ( inv g `  g ) `  x
) ( +g  `  g
) y )  e.  s ) } )
254, 4xpex 4801 . . . . 5  |-  ( X  X.  X )  e. 
_V
26 simpl 443 . . . . . . . 8  |-  ( ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S )  ->  { x ,  y }  C_  X
)
27 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
28 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
2927, 28prss 3769 . . . . . . . 8  |-  ( ( x  e.  X  /\  y  e.  X )  <->  { x ,  y } 
C_  X )
3026, 29sylibr 203 . . . . . . 7  |-  ( ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S )  ->  ( x  e.  X  /\  y  e.  X ) )
3130ssopab2i 4292 . . . . . 6  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  C_  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  X ) }
32 df-xp 4695 . . . . . 6  |-  ( X  X.  X )  =  { <. x ,  y
>.  |  ( x  e.  X  /\  y  e.  X ) }
3331, 32sseqtr4i 3211 . . . . 5  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  C_  ( X  X.  X )
3425, 33ssexi 4159 . . . 4  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  e.  _V
3523, 24, 34ovmpt2a 5978 . . 3  |-  ( ( G  e.  _V  /\  S  e.  _V )  ->  ( G ~QG  S )  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `
 x )  .+  y )  e.  S
) } )
366, 35syl5eq 2327 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } )
371, 5, 36syl2an 463 1  |-  ( ( G  e.  V  /\  S  C_  X )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {cpr 3641   {copab 4076    X. cxp 4687   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   inv gcminusg 14363   ~QG cqg 14617
This theorem is referenced by:  eqgval  14666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-eqg 14620
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