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Theorem eqgabl 16340
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 2443 . . 3  |-  ( invg `  G )  =  ( invg `  G )
3 eqid 2443 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqgabl.r . . 3  |-  .~  =  ( G ~QG  S )
51, 2, 3, 4eqgval 15751 . 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 16303 . . . . . . . . 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 15604 . . . . . . . 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 16315 . . . . . . 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 1218 . . . . . 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 15602 . . . . . . 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 2478 . . . . 5  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( invg `  G ) `  A
) ( +g  `  G
) B )  =  ( B  .-  A
) )
1918eleq1d 2509 . . . 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 967 . . 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 967 . . 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 965    = wceq 1369    e. wcel 1756    C_ wss 3349   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259   Grpcgrp 15431   invgcminusg 15432   -gcsg 15434   ~QG cqg 15698   Abelcabel 16299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-eqg 15701  df-cmn 16300  df-abl 16301
This theorem is referenced by:  2idlcpbl  17338  zndvds  18004  tgptsmscls  19746
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