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Theorem releqgg 13011
Description: The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypothesis
Ref Expression
releqg.r  |-  R  =  ( G ~QG  S )
Assertion
Ref Expression
releqgg  |-  ( ( G  e.  V  /\  S  e.  W )  ->  Rel  R )

Proof of Theorem releqgg
Dummy variables  i  r  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4752 . 2  |-  Rel  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  G )  /\  ( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  S
) }
2 releqg.r . . . 4  |-  R  =  ( G ~QG  S )
3 elex 2748 . . . . . 6  |-  ( G  e.  V  ->  G  e.  _V )
43adantr 276 . . . . 5  |-  ( ( G  e.  V  /\  S  e.  W )  ->  G  e.  _V )
5 elex 2748 . . . . . 6  |-  ( S  e.  W  ->  S  e.  _V )
65adantl 277 . . . . 5  |-  ( ( G  e.  V  /\  S  e.  W )  ->  S  e.  _V )
7 vex 2740 . . . . . . . . 9  |-  x  e. 
_V
8 vex 2740 . . . . . . . . 9  |-  y  e. 
_V
97, 8prss 3748 . . . . . . . 8  |-  ( ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  <->  { x ,  y }  C_  ( Base `  G )
)
109anbi1i 458 . . . . . . 7  |-  ( ( ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G ) )  /\  ( ( ( invg `  G
) `  x )
( +g  `  G ) y )  e.  S
)  <->  ( { x ,  y }  C_  ( Base `  G )  /\  ( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  S
) )
1110opabbii 4069 . . . . . 6  |-  { <. x ,  y >.  |  ( ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G ) )  /\  ( ( ( invg `  G
) `  x )
( +g  `  G ) y )  e.  S
) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  G )  /\  ( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  S
) }
12 basfn 12512 . . . . . . . . 9  |-  Base  Fn  _V
13 funfvex 5531 . . . . . . . . . 10  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
1413funfni 5315 . . . . . . . . 9  |-  ( (
Base  Fn  _V  /\  G  e.  _V )  ->  ( Base `  G )  e. 
_V )
1512, 4, 14sylancr 414 . . . . . . . 8  |-  ( ( G  e.  V  /\  S  e.  W )  ->  ( Base `  G
)  e.  _V )
16 xpexg 4739 . . . . . . . 8  |-  ( ( ( Base `  G
)  e.  _V  /\  ( Base `  G )  e.  _V )  ->  (
( Base `  G )  X.  ( Base `  G
) )  e.  _V )
1715, 15, 16syl2anc 411 . . . . . . 7  |-  ( ( G  e.  V  /\  S  e.  W )  ->  ( ( Base `  G
)  X.  ( Base `  G ) )  e. 
_V )
18 opabssxp 4699 . . . . . . . 8  |-  { <. x ,  y >.  |  ( ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G ) )  /\  ( ( ( invg `  G
) `  x )
( +g  `  G ) y )  e.  S
) }  C_  (
( Base `  G )  X.  ( Base `  G
) )
1918a1i 9 . . . . . . 7  |-  ( ( G  e.  V  /\  S  e.  W )  ->  { <. x ,  y
>.  |  ( (
x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  /\  (
( ( invg `  G ) `  x
) ( +g  `  G
) y )  e.  S ) }  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
2017, 19ssexd 4142 . . . . . 6  |-  ( ( G  e.  V  /\  S  e.  W )  ->  { <. x ,  y
>.  |  ( (
x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  /\  (
( ( invg `  G ) `  x
) ( +g  `  G
) y )  e.  S ) }  e.  _V )
2111, 20eqeltrrid 2265 . . . . 5  |-  ( ( G  e.  V  /\  S  e.  W )  ->  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( Base `  G
)  /\  ( (
( invg `  G ) `  x
) ( +g  `  G
) y )  e.  S ) }  e.  _V )
22 fveq2 5514 . . . . . . . . 9  |-  ( r  =  G  ->  ( Base `  r )  =  ( Base `  G
) )
2322sseq2d 3185 . . . . . . . 8  |-  ( r  =  G  ->  ( { x ,  y }  C_  ( Base `  r )  <->  { x ,  y }  C_  ( Base `  G )
) )
24 fveq2 5514 . . . . . . . . . 10  |-  ( r  =  G  ->  ( +g  `  r )  =  ( +g  `  G
) )
25 fveq2 5514 . . . . . . . . . . 11  |-  ( r  =  G  ->  ( invg `  r )  =  ( invg `  G ) )
2625fveq1d 5516 . . . . . . . . . 10  |-  ( r  =  G  ->  (
( invg `  r ) `  x
)  =  ( ( invg `  G
) `  x )
)
27 eqidd 2178 . . . . . . . . . 10  |-  ( r  =  G  ->  y  =  y )
2824, 26, 27oveq123d 5893 . . . . . . . . 9  |-  ( r  =  G  ->  (
( ( invg `  r ) `  x
) ( +g  `  r
) y )  =  ( ( ( invg `  G ) `
 x ) ( +g  `  G ) y ) )
2928eleq1d 2246 . . . . . . . 8  |-  ( r  =  G  ->  (
( ( ( invg `  r ) `
 x ) ( +g  `  r ) y )  e.  i  <-> 
( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  i ) )
3023, 29anbi12d 473 . . . . . . 7  |-  ( r  =  G  ->  (
( { x ,  y }  C_  ( Base `  r )  /\  ( ( ( invg `  r ) `
 x ) ( +g  `  r ) y )  e.  i )  <->  ( { x ,  y }  C_  ( Base `  G )  /\  ( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  i ) ) )
3130opabbidv 4068 . . . . . 6  |-  ( r  =  G  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  r )  /\  (
( ( invg `  r ) `  x
) ( +g  `  r
) y )  e.  i ) }  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( Base `  G
)  /\  ( (
( invg `  G ) `  x
) ( +g  `  G
) y )  e.  i ) } )
32 eleq2 2241 . . . . . . . 8  |-  ( i  =  S  ->  (
( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  i  <-> 
( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  S
) )
3332anbi2d 464 . . . . . . 7  |-  ( i  =  S  ->  (
( { x ,  y }  C_  ( Base `  G )  /\  ( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  i )  <->  ( { x ,  y }  C_  ( Base `  G )  /\  ( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  S
) ) )
3433opabbidv 4068 . . . . . 6  |-  ( i  =  S  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  G )  /\  (
( ( invg `  G ) `  x
) ( +g  `  G
) y )  e.  i ) }  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( Base `  G
)  /\  ( (
( invg `  G ) `  x
) ( +g  `  G
) y )  e.  S ) } )
35 df-eqg 12963 . . . . . 6  |- ~QG  =  ( r  e.  _V ,  i  e. 
_V  |->  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  r )  /\  (
( ( invg `  r ) `  x
) ( +g  `  r
) y )  e.  i ) } )
3631, 34, 35ovmpog 6006 . . . . 5  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  G )  /\  ( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  S
) }  e.  _V )  ->  ( G ~QG  S )  =  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  G )  /\  (
( ( invg `  G ) `  x
) ( +g  `  G
) y )  e.  S ) } )
374, 6, 21, 36syl3anc 1238 . . . 4  |-  ( ( G  e.  V  /\  S  e.  W )  ->  ( G ~QG  S )  =  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  G )  /\  ( ( ( invg `  G ) `
 x ) ( +g  `  G ) y )  e.  S
) } )
382, 37eqtrid 2222 . . 3  |-  ( ( G  e.  V  /\  S  e.  W )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  G )  /\  (
( ( invg `  G ) `  x
) ( +g  `  G
) y )  e.  S ) } )
3938releqd 4709 . 2  |-  ( ( G  e.  V  /\  S  e.  W )  ->  ( Rel  R  <->  Rel  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  G )  /\  (
( ( invg `  G ) `  x
) ( +g  `  G
) y )  e.  S ) } ) )
401, 39mpbiri 168 1  |-  ( ( G  e.  V  /\  S  e.  W )  ->  Rel  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129   {cpr 3593   {copab 4062    X. cxp 4623   Rel wrel 4630    Fn wfn 5210   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   invgcminusg 12810   ~QG cqg 12960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-inn 8916  df-ndx 12457  df-slot 12458  df-base 12460  df-eqg 12963
This theorem is referenced by:  eqger  13014  eqgid  13016
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