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Theorem eqgfval 16121
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  =  ( invg `  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 3127 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 eqgval.x . . . 4  |-  X  =  ( Base `  G
)
3 fvex 5882 . . . 4  |-  ( Base `  G )  e.  _V
42, 3eqeltri 2551 . . 3  |-  X  e. 
_V
54ssex 4597 . 2  |-  ( S 
C_  X  ->  S  e.  _V )
6 eqgval.r . . 3  |-  R  =  ( G ~QG  S )
7 simpl 457 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  g  =  G )
87fveq2d 5876 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( Base `  g
)  =  ( Base `  G ) )
98, 2syl6eqr 2526 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  ( Base `  g
)  =  X )
109sseq2d 3537 . . . . . 6  |-  ( ( g  =  G  /\  s  =  S )  ->  ( { x ,  y }  C_  ( Base `  g )  <->  { x ,  y }  C_  X ) )
117fveq2d 5876 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  ( +g  `  g
)  =  ( +g  `  G ) )
12 eqgval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
1311, 12syl6eqr 2526 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( +g  `  g
)  =  .+  )
147fveq2d 5876 . . . . . . . . . 10  |-  ( ( g  =  G  /\  s  =  S )  ->  ( invg `  g )  =  ( invg `  G
) )
15 eqgval.n . . . . . . . . . 10  |-  N  =  ( invg `  G )
1614, 15syl6eqr 2526 . . . . . . . . 9  |-  ( ( g  =  G  /\  s  =  S )  ->  ( invg `  g )  =  N )
1716fveq1d 5874 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( invg `  g ) `  x
)  =  ( N `
 x ) )
18 eqidd 2468 . . . . . . . 8  |-  ( ( g  =  G  /\  s  =  S )  ->  y  =  y )
1913, 17, 18oveq123d 6316 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( ( invg `  g ) `
 x ) ( +g  `  g ) y )  =  ( ( N `  x
)  .+  y )
)
20 simpr 461 . . . . . . 7  |-  ( ( g  =  G  /\  s  =  S )  ->  s  =  S )
2119, 20eleq12d 2549 . . . . . 6  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( ( ( invg `  g
) `  x )
( +g  `  g ) y )  e.  s  <-> 
( ( N `  x )  .+  y
)  e.  S ) )
2210, 21anbi12d 710 . . . . 5  |-  ( ( g  =  G  /\  s  =  S )  ->  ( ( { x ,  y }  C_  ( Base `  g )  /\  ( ( ( invg `  g ) `
 x ) ( +g  `  g ) y )  e.  s )  <->  ( { x ,  y }  C_  X  /\  ( ( N `
 x )  .+  y )  e.  S
) ) )
2322opabbidv 4516 . . . 4  |-  ( ( g  =  G  /\  s  =  S )  ->  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( Base `  g
)  /\  ( (
( invg `  g ) `  x
) ( +g  `  g
) y )  e.  s ) }  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  X  /\  (
( N `  x
)  .+  y )  e.  S ) } )
24 df-eqg 16072 . . . 4  |- ~QG  =  ( g  e.  _V ,  s  e. 
_V  |->  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  g )  /\  (
( ( invg `  g ) `  x
) ( +g  `  g
) y )  e.  s ) } )
254, 4xpex 6599 . . . . 5  |-  ( X  X.  X )  e. 
_V
26 simpl 457 . . . . . . . 8  |-  ( ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S )  ->  { x ,  y }  C_  X
)
27 vex 3121 . . . . . . . . 9  |-  x  e. 
_V
28 vex 3121 . . . . . . . . 9  |-  y  e. 
_V
2927, 28prss 4187 . . . . . . . 8  |-  ( ( x  e.  X  /\  y  e.  X )  <->  { x ,  y } 
C_  X )
3026, 29sylibr 212 . . . . . . 7  |-  ( ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S )  ->  ( x  e.  X  /\  y  e.  X ) )
3130ssopab2i 4781 . . . . . 6  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  C_  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  X ) }
32 df-xp 5011 . . . . . 6  |-  ( X  X.  X )  =  { <. x ,  y
>.  |  ( x  e.  X  /\  y  e.  X ) }
3331, 32sseqtr4i 3542 . . . . 5  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  C_  ( X  X.  X )
3425, 33ssexi 4598 . . . 4  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  e.  _V
3523, 24, 34ovmpt2a 6428 . . 3  |-  ( ( G  e.  _V  /\  S  e.  _V )  ->  ( G ~QG  S )  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `
 x )  .+  y )  e.  S
) } )
366, 35syl5eq 2520 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } )
371, 5, 36syl2an 477 1  |-  ( ( G  e.  V  /\  S  C_  X )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   {cpr 4035   {copab 4510    X. cxp 5003   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   invgcminusg 15926   ~QG cqg 16069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-eqg 16072
This theorem is referenced by:  eqgval  16122
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