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Theorem eqgval 16229
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  =  ( invg `  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  =  ( invg `  G )
3 eqgval.p . . . 4  |-  .+  =  ( +g  `  G )
4 eqgval.r . . . 4  |-  R  =  ( G ~QG  S )
51, 2, 3, 4eqgfval 16228 . . 3  |-  ( ( G  e.  V  /\  S  C_  X )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y
)  e.  S ) } )
65breqd 4448 . 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 6327 . . . 4  |-  ( A { <. x ,  y
>.  |  ( {
x ,  y } 
C_  X  /\  (
( N `  x
)  .+  y )  e.  S ) } B  ->  ( A  e.  _V  /\  B  e.  _V )
)
87adantl 466 . . 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 1003 . . . . 5  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  A  e.  X )
10 elex 3104 . . . . 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 1004 . . . . 5  |-  ( ( ( G  e.  V  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A ) 
.+  B )  e.  S ) )  ->  B  e.  X )
13 elex 3104 . . . . 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 532 . . 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 3098 . . . . . . . 8  |-  x  e. 
_V
17 vex 3098 . . . . . . . 8  |-  y  e. 
_V
1816, 17prss 4169 . . . . . . 7  |-  ( ( x  e.  X  /\  y  e.  X )  <->  { x ,  y } 
C_  X )
19 eleq1 2515 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  X  <->  A  e.  X ) )
20 eleq1 2515 . . . . . . . 8  |-  ( y  =  B  ->  (
y  e.  X  <->  B  e.  X ) )
2119, 20bi2anan9 873 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  X  /\  y  e.  X )  <->  ( A  e.  X  /\  B  e.  X ) ) )
2218, 21syl5bbr 259 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( { x ,  y }  C_  X  <->  ( A  e.  X  /\  B  e.  X )
) )
23 fveq2 5856 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
24 id 22 . . . . . . . 8  |-  ( y  =  B  ->  y  =  B )
2523, 24oveqan12d 6300 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( N `  x )  .+  y
)  =  ( ( N `  A ) 
.+  B ) )
2625eleq1d 2512 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( N `
 x )  .+  y )  e.  S  <->  ( ( N `  A
)  .+  B )  e.  S ) )
2722, 26anbi12d 710 . . . . 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 976 . . . . 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 263 . . . 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 2443 . . . 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 4751 . . 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 900 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   {cpr 4016   class class class wbr 4437   {copab 4494   ` cfv 5578  (class class class)co 6281   Basecbs 14614   +g cplusg 14679   invgcminusg 16033   ~QG cqg 16176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-eqg 16179
This theorem is referenced by:  eqger  16230  eqglact  16231  eqgid  16232  eqgcpbl  16234  gastacos  16327  orbstafun  16328  sylow2blem1  16619  sylow2blem3  16621  eqgabl  16822  tgpconcompeqg  20588  tgpconcomp  20589  qustgpopn  20596
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