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Theorem grpinvsub 15992
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  =  ( invg `  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  =  ( invg `  G )
31, 2grpinvcl 15967 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
433adant2 1015 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
5 eqid 2467 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
61, 5, 2grpinvadd 15988 . . . 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 1273 . . 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 15977 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
983adant2 1015 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
109oveq1d 6310 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  ( N `  Y ) ) ( +g  `  G
) ( N `  X ) )  =  ( Y ( +g  `  G ) ( N `
 X ) ) )
117, 10eqtrd 2508 . 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 15965 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
14133adant1 1014 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
1514fveq2d 5876 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
161, 5, 2, 12grpsubval 15965 . . . 4  |-  ( ( Y  e.  B  /\  X  e.  B )  ->  ( Y  .-  X
)  =  ( Y ( +g  `  G
) ( N `  X ) ) )
1716ancoms 453 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( Y  .-  X
)  =  ( Y ( +g  `  G
) ( N `  X ) ) )
18173adant1 1014 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .-  X
)  =  ( Y ( +g  `  G
) ( N `  X ) ) )
1911, 15, 183eqtr4d 2518 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   Grpcgrp 15925   invgcminusg 15926   -gcsg 15927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931
This theorem is referenced by:  grpsubsub  15999  ablsub2inv  16694  lspsnsub  17524  ghmcnp  20481  nrmmetd  20963  nmsub  21010  mapdpglem14  36883
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