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Theorem eqgabl 16716
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 2467 . . 3  |-  ( invg `  G )  =  ( invg `  G )
3 eqid 2467 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqgabl.r . . 3  |-  .~  =  ( G ~QG  S )
51, 2, 3, 4eqgval 16122 . 2  |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( ( ( invg `  G ) `  A
) ( +g  `  G
) B )  e.  S ) ) )
6 simpll 753 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  G  e.  Abel )
7 ablgrp 16676 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
87ad2antrr 725 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  G  e.  Grp )
9 simprl 755 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  A  e.  X )
101, 2grpinvcl 15967 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
118, 9, 10syl2anc 661 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( invg `  G ) `  A
)  e.  X )
12 simprr 756 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  B  e.  X )
131, 3ablcom 16688 . . . . . . 7  |-  ( ( G  e.  Abel  /\  (
( invg `  G ) `  A
)  e.  X  /\  B  e.  X )  ->  ( ( ( invg `  G ) `
 A ) ( +g  `  G ) B )  =  ( B ( +g  `  G
) ( ( invg `  G ) `
 A ) ) )
146, 11, 12, 13syl3anc 1228 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( invg `  G ) `  A
) ( +g  `  G
) B )  =  ( B ( +g  `  G ) ( ( invg `  G
) `  A )
) )
15 eqgabl.n . . . . . . . 8  |-  .-  =  ( -g `  G )
161, 3, 2, 15grpsubval 15965 . . . . . . 7  |-  ( ( B  e.  X  /\  A  e.  X )  ->  ( B  .-  A
)  =  ( B ( +g  `  G
) ( ( invg `  G ) `
 A ) ) )
1712, 9, 16syl2anc 661 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( B  .-  A )  =  ( B ( +g  `  G ) ( ( invg `  G
) `  A )
) )
1814, 17eqtr4d 2511 . . . . 5  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( invg `  G ) `  A
) ( +g  `  G
) B )  =  ( B  .-  A
) )
1918eleq1d 2536 . . . 4  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( ( invg `  G ) `
 A ) ( +g  `  G ) B )  e.  S  <->  ( B  .-  A )  e.  S ) )
2019pm5.32da 641 . . 3  |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  (
( ( A  e.  X  /\  B  e.  X )  /\  (
( ( invg `  G ) `  A
) ( +g  `  G
) B )  e.  S )  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( B  .-  A
)  e.  S ) ) )
21 df-3an 975 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  ( ( ( invg `  G ) `
 A ) ( +g  `  G ) B )  e.  S
)  <->  ( ( A  e.  X  /\  B  e.  X )  /\  (
( ( invg `  G ) `  A
) ( +g  `  G
) B )  e.  S ) )
22 df-3an 975 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  ( B  .-  A )  e.  S )  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( B  .-  A
)  e.  S ) )
2320, 21, 223bitr4g 288 . 2  |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  (
( A  e.  X  /\  B  e.  X  /\  ( ( ( invg `  G ) `
 A ) ( +g  `  G ) B )  e.  S
)  <->  ( A  e.  X  /\  B  e.  X  /\  ( B 
.-  A )  e.  S ) ) )
245, 23bitrd 253 1  |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( B 
.-  A )  e.  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3481   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   Grpcgrp 15925   invgcminusg 15926   -gcsg 15927   ~QG cqg 16069   Abelcabl 16672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-eqg 16072  df-cmn 16673  df-abl 16674
This theorem is referenced by:  2idlcpbl  17752  zndvds  18457  tgptsmscls  20520
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