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Theorem eqgfval 14988
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 2964 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 eqgval.x . . . 4  |-  X  =  ( Base `  G
)
3 fvex 5742 . . . 4  |-  ( Base `  G )  e.  _V
42, 3eqeltri 2506 . . 3  |-  X  e. 
_V
54ssex 4347 . 2  |-  ( S 
C_  X  ->  S  e.  _V )
6 eqgval.r . . 3  |-  R  =  ( G ~QG  S )
7 simpl 444 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  g  =  G )
87fveq2d 5732 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( Base `  g
)  =  ( Base `  G ) )
98, 2syl6eqr 2486 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  ( Base `  g
)  =  X )
109sseq2d 3376 . . . . . 6  |-  ( ( g  =  G  /\  s  =  S )  ->  ( { x ,  y }  C_  ( Base `  g )  <->  { x ,  y }  C_  X ) )
117fveq2d 5732 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  ( +g  `  g
)  =  ( +g  `  G ) )
12 eqgval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
1311, 12syl6eqr 2486 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( +g  `  g
)  =  .+  )
147fveq2d 5732 . . . . . . . . . 10  |-  ( ( g  =  G  /\  s  =  S )  ->  ( inv g `  g )  =  ( inv g `  G
) )
15 eqgval.n . . . . . . . . . 10  |-  N  =  ( inv g `  G )
1614, 15syl6eqr 2486 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  ( inv g `  g )  =  N )
1716fveq1d 5730 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( inv g `  g ) `  x
)  =  ( N `
 x ) )
18 eqidd 2437 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  y  =  y )
1913, 17, 18oveq123d 6102 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( ( inv g `  g ) `
 x ) ( +g  `  g ) y )  =  ( ( N `  x
)  .+  y )
)
20 simpr 448 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  s  =  S )
2119, 20eleq12d 2504 . . . . . 6  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( ( ( inv g `  g
) `  x )
( +g  `  g ) y )  e.  s  <-> 
( ( N `  x )  .+  y
)  e.  S ) )
2210, 21anbi12d 692 . . . . 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 4271 . . . 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 14943 . . . 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 4990 . . . . 5  |-  ( X  X.  X )  e. 
_V
26 simpl 444 . . . . . . . 8  |-  ( ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S )  ->  { x ,  y }  C_  X
)
27 vex 2959 . . . . . . . . 9  |-  x  e. 
_V
28 vex 2959 . . . . . . . . 9  |-  y  e. 
_V
2927, 28prss 3952 . . . . . . . 8  |-  ( ( x  e.  X  /\  y  e.  X )  <->  { x ,  y } 
C_  X )
3026, 29sylibr 204 . . . . . . 7  |-  ( ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S )  ->  ( x  e.  X  /\  y  e.  X ) )
3130ssopab2i 4482 . . . . . 6  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  C_  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  X ) }
32 df-xp 4884 . . . . . 6  |-  ( X  X.  X )  =  { <. x ,  y
>.  |  ( x  e.  X  /\  y  e.  X ) }
3331, 32sseqtr4i 3381 . . . . 5  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  C_  ( X  X.  X )
3425, 33ssexi 4348 . . . 4  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  e.  _V
3523, 24, 34ovmpt2a 6204 . . 3  |-  ( ( G  e.  _V  /\  S  e.  _V )  ->  ( G ~QG  S )  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `
 x )  .+  y )  e.  S
) } )
366, 35syl5eq 2480 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } )
371, 5, 36syl2an 464 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   {cpr 3815   {copab 4265    X. cxp 4876   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   inv gcminusg 14686   ~QG cqg 14940
This theorem is referenced by:  eqgval  14989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-eqg 14943
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