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Theorem eqgfval 14764
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 2872 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 eqgval.x . . . 4  |-  X  =  ( Base `  G
)
3 fvex 5622 . . . 4  |-  ( Base `  G )  e.  _V
42, 3eqeltri 2428 . . 3  |-  X  e. 
_V
54ssex 4239 . 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 5612 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( Base `  g
)  =  ( Base `  G ) )
98, 2syl6eqr 2408 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  ( Base `  g
)  =  X )
109sseq2d 3282 . . . . . 6  |-  ( ( g  =  G  /\  s  =  S )  ->  ( { x ,  y }  C_  ( Base `  g )  <->  { x ,  y }  C_  X ) )
117fveq2d 5612 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  ( +g  `  g
)  =  ( +g  `  G ) )
12 eqgval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
1311, 12syl6eqr 2408 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( +g  `  g
)  =  .+  )
147fveq2d 5612 . . . . . . . . . 10  |-  ( ( g  =  G  /\  s  =  S )  ->  ( inv g `  g )  =  ( inv g `  G
) )
15 eqgval.n . . . . . . . . . 10  |-  N  =  ( inv g `  G )
1614, 15syl6eqr 2408 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  ( inv g `  g )  =  N )
1716fveq1d 5610 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( inv g `  g ) `  x
)  =  ( N `
 x ) )
18 eqidd 2359 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  y  =  y )
1913, 17, 18oveq123d 5966 . . . . . . 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 2426 . . . . . 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 4163 . . . 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 14719 . . . 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 4883 . . . . 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 2867 . . . . . . . . 9  |-  x  e. 
_V
28 vex 2867 . . . . . . . . 9  |-  y  e. 
_V
2927, 28prss 3848 . . . . . . . 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 4374 . . . . . 6  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  C_  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  X ) }
32 df-xp 4777 . . . . . 6  |-  ( X  X.  X )  =  { <. x ,  y
>.  |  ( x  e.  X  /\  y  e.  X ) }
3331, 32sseqtr4i 3287 . . . . 5  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  C_  ( X  X.  X )
3425, 33ssexi 4240 . . . 4  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  e.  _V
3523, 24, 34ovmpt2a 6065 . . 3  |-  ( ( G  e.  _V  /\  S  e.  _V )  ->  ( G ~QG  S )  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `
 x )  .+  y )  e.  S
) } )
366, 35syl5eq 2402 . 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 1642    e. wcel 1710   _Vcvv 2864    C_ wss 3228   {cpr 3717   {copab 4157    X. cxp 4769   ` cfv 5337  (class class class)co 5945   Basecbs 13245   +g cplusg 13305   inv gcminusg 14462   ~QG cqg 14716
This theorem is referenced by:  eqgval  14765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-eqg 14719
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