MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqgabl Structured version   Unicode version

Theorem eqgabl 15446
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 2435 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
3 eqid 2435 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqgabl.r . . 3  |-  .~  =  ( G ~QG  S )
51, 2, 3, 4eqgval 14981 . 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 15409 . . . . . . . . 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 14842 . . . . . . . 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 15421 . . . . . . 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 14840 . . . . . . 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 2470 . . . . 5  |-  ( ( ( G  e.  Abel  /\  S  C_  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( inv g `  G ) `  A
) ( +g  `  G
) B )  =  ( B  .-  A
) )
1918eleq1d 2501 . . . 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 1652    e. wcel 1725    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677   inv gcminusg 14678   -gcsg 14680   ~QG cqg 14932   Abelcabel 15405
This theorem is referenced by:  2idlcpbl  16297  zndvds  16822  tgptsmscls  18171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-eqg 14935  df-cmn 15406  df-abl 15407
  Copyright terms: Public domain W3C validator