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Theorem grpinvsub 14859
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 14838 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
433adant2 976 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
5 eqid 2435 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
61, 5, 2grpinvadd 14855 . . . 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 1229 . . 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 14846 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
983adant2 976 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
109oveq1d 6087 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  ( N `  Y ) ) ( +g  `  G
) ( N `  X ) )  =  ( Y ( +g  `  G ) ( N `
 X ) ) )
117, 10eqtrd 2467 . 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 14836 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
14133adant1 975 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
1514fveq2d 5723 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
161, 5, 2, 12grpsubval 14836 . . . 4  |-  ( ( Y  e.  B  /\  X  e.  B )  ->  ( Y  .-  X
)  =  ( Y ( +g  `  G
) ( N `  X ) ) )
1716ancoms 440 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( Y  .-  X
)  =  ( Y ( +g  `  G
) ( N `  X ) ) )
18173adant1 975 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .-  X
)  =  ( Y ( +g  `  G
) ( N `  X ) ) )
1911, 15, 183eqtr4d 2477 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 936    = wceq 1652    e. wcel 1725   ` cfv 5445  (class class class)co 6072   Basecbs 13457   +g cplusg 13517   Grpcgrp 14673   inv gcminusg 14674   -gcsg 14676
This theorem is referenced by:  grpsubsub  14865  ablsub2inv  15423  lspsnsub  16071  ghmcnp  18132  nrmmetd  18610  nmsub  18657  mapdpglem14  32322
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 4692
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-0g 13715  df-mnd 14678  df-grp 14800  df-minusg 14801  df-sbg 14802
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