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Theorem eqgen 13638
Description: Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
eqger.x  |-  X  =  ( Base `  G
)
eqger.r  |-  .~  =  ( G ~QG  Y )
Assertion
Ref Expression
eqgen  |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  ( X /.  .~  ) )  ->  Y  ~~  A )

Proof of Theorem eqgen
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2206 . 2  |-  ( X /.  .~  )  =  ( X /.  .~  )
2 breq2 4055 . 2  |-  ( [ x ]  .~  =  A  ->  ( Y  ~~  [ x ]  .~  <->  Y  ~~  A ) )
3 simpl 109 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  e.  (SubGrp `  G )
)
4 subgrcl 13590 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5 eqger.x . . . . . . . 8  |-  X  =  ( Base `  G
)
65subgss 13585 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
74, 6jca 306 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  ( G  e.  Grp  /\  Y  C_  X ) )
8 eqger.r . . . . . . . 8  |-  .~  =  ( G ~QG  Y )
9 eqid 2206 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
105, 8, 9eqglact 13636 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  x  e.  X )  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
11103expa 1206 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  x  e.  X
)  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
127, 11sylan 283 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
135, 8eqger 13635 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  .~  Er  X
)
14 basfn 12965 . . . . . . . . . 10  |-  Base  Fn  _V
154elexd 2787 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  _V )
16 funfvex 5606 . . . . . . . . . . 11  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
1716funfni 5385 . . . . . . . . . 10  |-  ( (
Base  Fn  _V  /\  G  e.  _V )  ->  ( Base `  G )  e. 
_V )
1814, 15, 17sylancr 414 . . . . . . . . 9  |-  ( Y  e.  (SubGrp `  G
)  ->  ( Base `  G )  e.  _V )
195, 18eqeltrid 2293 . . . . . . . 8  |-  ( Y  e.  (SubGrp `  G
)  ->  X  e.  _V )
20 erex 6657 . . . . . . . 8  |-  (  .~  Er  X  ->  ( X  e.  _V  ->  .~  e.  _V ) )
2113, 19, 20sylc 62 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  .~  e.  _V )
22 ecexg 6637 . . . . . . 7  |-  (  .~  e.  _V  ->  [ x ]  .~  e.  _V )
2321, 22syl 14 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  [ x ]  .~  e.  _V )
2423adantr 276 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  [ x ]  .~  e.  _V )
2512, 24eqeltrrd 2284 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y )  e.  _V )
26 eqid 2206 . . . . . . . . 9  |-  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )  =  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )
2726, 5, 9grplactf1o 13510 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G ) z ) ) ) `
 x ) : X -1-1-onto-> X )
2826, 5grplactfval 13508 . . . . . . . . . 10  |-  ( x  e.  X  ->  (
( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G ) z ) ) ) `  x
)  =  ( z  e.  X  |->  ( x ( +g  `  G
) z ) ) )
2928adantl 277 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G ) z ) ) ) `
 x )  =  ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) )
3029f1oeq1d 5529 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) ) `  x ) : X -1-1-onto-> X  <->  ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) : X -1-1-onto-> X ) )
3127, 30mpbid 147 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) : X -1-1-onto-> X )
324, 31sylan 283 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-onto-> X )
33 f1of1 5533 . . . . . 6  |-  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-onto-> X  ->  ( z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-> X )
3432, 33syl 14 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-> X )
356adantr 276 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  C_  X )
36 f1ores 5549 . . . . 5  |-  ( ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) : X -1-1-> X  /\  Y  C_  X )  ->  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )  |`  Y ) : Y -1-1-onto-> (
( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )
3734, 35, 36syl2anc 411 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) )  |`  Y ) : Y -1-1-onto-> ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )
38 f1oen2g 6859 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y )  e.  _V  /\  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) )  |`  Y ) : Y -1-1-onto-> ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )  ->  Y  ~~  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
393, 25, 37, 38syl3anc 1250 . . 3  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  ~~  ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )
4039, 12breqtrrd 4079 . 2  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  ~~  [ x ]  .~  )
411, 2, 40ectocld 6701 1  |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  ( X /.  .~  ) )  ->  Y  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1373    e. wcel 2177   _Vcvv 2773    C_ wss 3170   class class class wbr 4051    |-> cmpt 4113    |` cres 4685   "cima 4686    Fn wfn 5275   -1-1->wf1 5277   -1-1-onto->wf1o 5279   ` cfv 5280  (class class class)co 5957    Er wer 6630   [cec 6631   /.cqs 6632    ~~ cen 6838   Basecbs 12907   +g cplusg 12984   Grpcgrp 13407  SubGrpcsubg 13578   ~QG cqg 13580
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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-ltadd 8061
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-er 6633  df-ec 6635  df-qs 6639  df-en 6841  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-iress 12915  df-plusg 12997  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-subg 13581  df-eqg 13583
This theorem is referenced by: (None)
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