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Theorem eqgval 14916
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 14915 . . 3  |-  ( ( G  e.  V  /\  S  C_  X )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } )
65breqd 4164 . 2  |-  ( ( G  e.  V  /\  S  C_  X )  -> 
( A R B  <-> 
A { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } B ) )
7 brabv 6059 . . . 4  |-  ( A { <. x ,  y
>.  |  ( {
x ,  y } 
C_  X  /\  (
( N `  x
)  .+  y )  e.  S ) } B  ->  ( A  e.  _V  /\  B  e.  _V )
)
87adantl 453 . . 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 )
)
9 simpr1 963 . . . . 5  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  A  e.  X )
10 elex 2907 . . . . 5  |-  ( A  e.  X  ->  A  e.  _V )
119, 10syl 16 . . . 4  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  A  e.  _V )
12 simpr2 964 . . . . 5  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  B  e.  X )
13 elex 2907 . . . . 5  |-  ( B  e.  X  ->  B  e.  _V )
1412, 13syl 16 . . . 4  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  B  e.  _V )
1511, 14jca 519 . . 3  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
16 vex 2902 . . . . . . . 8  |-  x  e. 
_V
17 vex 2902 . . . . . . . 8  |-  y  e. 
_V
1816, 17prss 3895 . . . . . . 7  |-  ( ( x  e.  X  /\  y  e.  X )  <->  { x ,  y } 
C_  X )
19 eleq1 2447 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  X  <->  A  e.  X ) )
20 eleq1 2447 . . . . . . . 8  |-  ( y  =  B  ->  (
y  e.  X  <->  B  e.  X ) )
2119, 20bi2anan9 844 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  X  /\  y  e.  X )  <->  ( A  e.  X  /\  B  e.  X ) ) )
2218, 21syl5bbr 251 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( { x ,  y }  C_  X  <->  ( A  e.  X  /\  B  e.  X )
) )
23 fveq2 5668 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
24 id 20 . . . . . . . 8  |-  ( y  =  B  ->  y  =  B )
2523, 24oveqan12d 6039 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( N `  x )  .+  y
)  =  ( ( N `  A ) 
.+  B ) )
2625eleq1d 2453 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( N `
 x )  .+  y )  e.  S  <->  ( ( N `  A
)  .+  B )  e.  S ) )
2722, 26anbi12d 692 . . . . 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 ) ) )
28 df-3an 938 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A )  .+  B
)  e.  S )  <-> 
( ( A  e.  X  /\  B  e.  X )  /\  (
( N `  A
)  .+  B )  e.  S ) )
2927, 28syl6bbr 255 . . . 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 ) ) )
30 eqid 2387 . . . 4  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) }
3129, 30brabga 4410 . . 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 ) ) )
328, 15, 31pm5.21nd 869 . 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 ) ) )
336, 32bitrd 245 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263   {cpr 3758   class class class wbr 4153   {copab 4206   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   inv gcminusg 14613   ~QG cqg 14867
This theorem is referenced by:  eqger  14917  eqglact  14918  eqgid  14919  eqgcpbl  14921  gastacos  15014  orbstafun  15015  sylow2blem1  15181  sylow2blem3  15183  eqgabl  15381  tgpconcompeqg  18062  tgpconcomp  18063  divstgpopn  18070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-eqg 14870
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