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Theorem eqgabl 15131
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 2283 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
3 eqid 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqgabl.r . . 3  |-  .~  =  ( G ~QG  S )
51, 2, 3, 4eqgval 14666 . 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 730 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  G  e.  Abel )
7 ablgrp 15094 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
87ad2antrr 706 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  G  e.  Grp )
9 simprl 732 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  A  e.  X )
101, 2grpinvcl 14527 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( inv g `  G ) `  A
)  e.  X )
118, 9, 10syl2anc 642 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( inv g `  G ) `  A
)  e.  X )
12 simprr 733 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  B  e.  X )
131, 3ablcom 15106 . . . . . . 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 1182 . . . . . 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 14525 . . . . . . 7  |-  ( ( B  e.  X  /\  A  e.  X )  ->  ( B  .-  A
)  =  ( B ( +g  `  G
) ( ( inv g `  G ) `
 A ) ) )
1712, 9, 16syl2anc 642 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( B  .-  A )  =  ( B ( +g  `  G ) ( ( inv g `  G
) `  A )
) )
1814, 17eqtr4d 2318 . . . . 5  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( inv g `  G ) `  A
) ( +g  `  G
) B )  =  ( B  .-  A
) )
1918eleq1d 2349 . . . 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 622 . . 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 936 . . 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 936 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  ( B  .-  A )  e.  S )  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( B  .-  A
)  e.  S ) )
2320, 21, 223bitr4g 279 . 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 244 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365   ~QG cqg 14617   Abelcabel 15090
This theorem is referenced by:  2idlcpbl  15986  zndvds  16503  tgptsmscls  17832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-eqg 14620  df-cmn 15091  df-abl 15092
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