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Theorem eqgabl 15382
Description: Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
eqgabl.x  |-  X  =  ( Base `  G
)
eqgabl.n  |-  .-  =  ( -g `  G )
eqgabl.r  |-  .~  =  ( G ~QG  S )
Assertion
Ref Expression
eqgabl  |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( B 
.-  A )  e.  S ) ) )

Proof of Theorem eqgabl
StepHypRef Expression
1 eqgabl.x . . 3  |-  X  =  ( Base `  G
)
2 eqid 2388 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
3 eqid 2388 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqgabl.r . . 3  |-  .~  =  ( G ~QG  S )
51, 2, 3, 4eqgval 14917 . 2  |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( ( ( inv g `  G ) `  A
) ( +g  `  G
) B )  e.  S ) ) )
6 simpll 731 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  G  e.  Abel )
7 ablgrp 15345 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
87ad2antrr 707 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  G  e.  Grp )
9 simprl 733 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  A  e.  X )
101, 2grpinvcl 14778 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( inv g `  G ) `  A
)  e.  X )
118, 9, 10syl2anc 643 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( inv g `  G ) `  A
)  e.  X )
12 simprr 734 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  B  e.  X )
131, 3ablcom 15357 . . . . . . 7  |-  ( ( G  e.  Abel  /\  (
( inv g `  G ) `  A
)  e.  X  /\  B  e.  X )  ->  ( ( ( inv g `  G ) `
 A ) ( +g  `  G ) B )  =  ( B ( +g  `  G
) ( ( inv g `  G ) `
 A ) ) )
146, 11, 12, 13syl3anc 1184 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( inv g `  G ) `  A
) ( +g  `  G
) B )  =  ( B ( +g  `  G ) ( ( inv g `  G
) `  A )
) )
15 eqgabl.n . . . . . . . 8  |-  .-  =  ( -g `  G )
161, 3, 2, 15grpsubval 14776 . . . . . . 7  |-  ( ( B  e.  X  /\  A  e.  X )  ->  ( B  .-  A
)  =  ( B ( +g  `  G
) ( ( inv g `  G ) `
 A ) ) )
1712, 9, 16syl2anc 643 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( B  .-  A )  =  ( B ( +g  `  G ) ( ( inv g `  G
) `  A )
) )
1814, 17eqtr4d 2423 . . . . 5  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( inv g `  G ) `  A
) ( +g  `  G
) B )  =  ( B  .-  A
) )
1918eleq1d 2454 . . . 4  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( ( inv g `  G ) `
 A ) ( +g  `  G ) B )  e.  S  <->  ( B  .-  A )  e.  S ) )
2019pm5.32da 623 . . 3  |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  (
( ( A  e.  X  /\  B  e.  X )  /\  (
( ( inv g `  G ) `  A
) ( +g  `  G
) B )  e.  S )  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( B  .-  A
)  e.  S ) ) )
21 df-3an 938 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  ( ( ( inv g `  G ) `
 A ) ( +g  `  G ) B )  e.  S
)  <->  ( ( A  e.  X  /\  B  e.  X )  /\  (
( ( inv g `  G ) `  A
) ( +g  `  G
) B )  e.  S ) )
22 df-3an 938 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  ( B  .-  A )  e.  S )  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( B  .-  A
)  e.  S ) )
2320, 21, 223bitr4g 280 . 2  |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  (
( A  e.  X  /\  B  e.  X  /\  ( ( ( inv g `  G ) `
 A ) ( +g  `  G ) B )  e.  S
)  <->  ( A  e.  X  /\  B  e.  X  /\  ( B 
.-  A )  e.  S ) ) )
245, 23bitrd 245 1  |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( B 
.-  A )  e.  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3264   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   Grpcgrp 14613   inv gcminusg 14614   -gcsg 14616   ~QG cqg 14868   Abelcabel 15341
This theorem is referenced by:  2idlcpbl  16233  zndvds  16754  tgptsmscls  18101
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-sbg 14742  df-eqg 14871  df-cmn 15342  df-abl 15343
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