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Theorem grpinvsub 14902
Description: Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)
Hypotheses
Ref Expression
grpsubcl.b  |-  B  =  ( Base `  G
)
grpsubcl.m  |-  .-  =  ( -g `  G )
grpinvsub.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
grpinvsub  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )

Proof of Theorem grpinvsub
StepHypRef Expression
1 grpsubcl.b . . . . . 6  |-  B  =  ( Base `  G
)
2 grpinvsub.n . . . . . 6  |-  N  =  ( inv g `  G )
31, 2grpinvcl 14881 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
433adant2 977 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
5 eqid 2442 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
61, 5, 2grpinvadd 14898 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( N `  Y )  e.  B )  -> 
( N `  ( X ( +g  `  G
) ( N `  Y ) ) )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
74, 6syld3an3 1230 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
81, 2grpinvinv 14889 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
983adant2 977 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
109oveq1d 6125 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  ( N `  Y ) ) ( +g  `  G
) ( N `  X ) )  =  ( Y ( +g  `  G ) ( N `
 X ) ) )
117, 10eqtrd 2474 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) )  =  ( Y ( +g  `  G ) ( N `  X
) ) )
12 grpsubcl.m . . . . 5  |-  .-  =  ( -g `  G )
131, 5, 2, 12grpsubval 14879 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
14133adant1 976 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
1514fveq2d 5761 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
161, 5, 2, 12grpsubval 14879 . . . 4  |-  ( ( Y  e.  B  /\  X  e.  B )  ->  ( Y  .-  X
)  =  ( Y ( +g  `  G
) ( N `  X ) ) )
1716ancoms 441 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( Y  .-  X
)  =  ( Y ( +g  `  G
) ( N `  X ) ) )
18173adant1 976 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .-  X
)  =  ( Y ( +g  `  G
) ( N `  X ) ) )
1911, 15, 183eqtr4d 2484 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1727   ` cfv 5483  (class class class)co 6110   Basecbs 13500   +g cplusg 13560   Grpcgrp 14716   inv gcminusg 14717   -gcsg 14719
This theorem is referenced by:  grpsubsub  14908  ablsub2inv  15466  lspsnsub  16114  ghmcnp  18175  nrmmetd  18653  nmsub  18700  mapdpglem14  32581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-0g 13758  df-mnd 14721  df-grp 14843  df-minusg 14844  df-sbg 14845
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