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Theorem ablsub2inv 16694
Description: Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
Hypotheses
Ref Expression
ablsub2inv.b  |-  B  =  ( Base `  G
)
ablsub2inv.m  |-  .-  =  ( -g `  G )
ablsub2inv.n  |-  N  =  ( invg `  G )
ablsub2inv.g  |-  ( ph  ->  G  e.  Abel )
ablsub2inv.x  |-  ( ph  ->  X  e.  B )
ablsub2inv.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
ablsub2inv  |-  ( ph  ->  ( ( N `  X )  .-  ( N `  Y )
)  =  ( Y 
.-  X ) )

Proof of Theorem ablsub2inv
StepHypRef Expression
1 ablsub2inv.b . . 3  |-  B  =  ( Base `  G
)
2 eqid 2467 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 ablsub2inv.m . . 3  |-  .-  =  ( -g `  G )
4 ablsub2inv.n . . 3  |-  N  =  ( invg `  G )
5 ablsub2inv.g . . . 4  |-  ( ph  ->  G  e.  Abel )
6 ablgrp 16676 . . . 4  |-  ( G  e.  Abel  ->  G  e. 
Grp )
75, 6syl 16 . . 3  |-  ( ph  ->  G  e.  Grp )
8 ablsub2inv.x . . . 4  |-  ( ph  ->  X  e.  B )
91, 4grpinvcl 15967 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
107, 8, 9syl2anc 661 . . 3  |-  ( ph  ->  ( N `  X
)  e.  B )
11 ablsub2inv.y . . 3  |-  ( ph  ->  Y  e.  B )
121, 2, 3, 4, 7, 10, 11grpsubinv 15983 . 2  |-  ( ph  ->  ( ( N `  X )  .-  ( N `  Y )
)  =  ( ( N `  X ) ( +g  `  G
) Y ) )
131, 2ablcom 16688 . . . . . 6  |-  ( ( G  e.  Abel  /\  ( N `  X )  e.  B  /\  Y  e.  B )  ->  (
( N `  X
) ( +g  `  G
) Y )  =  ( Y ( +g  `  G ) ( N `
 X ) ) )
145, 10, 11, 13syl3anc 1228 . . . . 5  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( Y ( +g  `  G ) ( N `  X
) ) )
151, 4grpinvinv 15977 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
167, 11, 15syl2anc 661 . . . . . 6  |-  ( ph  ->  ( N `  ( N `  Y )
)  =  Y )
1716oveq1d 6310 . . . . 5  |-  ( ph  ->  ( ( N `  ( N `  Y ) ) ( +g  `  G
) ( N `  X ) )  =  ( Y ( +g  `  G ) ( N `
 X ) ) )
1814, 17eqtr4d 2511 . . . 4  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
191, 4grpinvcl 15967 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
207, 11, 19syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  Y
)  e.  B )
211, 2, 4grpinvadd 15988 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( N `  Y )  e.  B )  -> 
( N `  ( X ( +g  `  G
) ( N `  Y ) ) )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
227, 8, 20, 21syl3anc 1228 . . . 4  |-  ( ph  ->  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
2318, 22eqtr4d 2511 . . 3  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
241, 2, 4, 3grpsubval 15965 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
258, 11, 24syl2anc 661 . . . 4  |-  ( ph  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
2625fveq2d 5876 . . 3  |-  ( ph  ->  ( N `  ( X  .-  Y ) )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
2723, 26eqtr4d 2511 . 2  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( N `  ( X  .-  Y ) ) )
281, 3, 4grpinvsub 15992 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
297, 8, 11, 28syl3anc 1228 . 2  |-  ( ph  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
3012, 27, 293eqtrd 2512 1  |-  ( ph  ->  ( ( N `  X )  .-  ( N `  Y )
)  =  ( Y 
.-  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   Grpcgrp 15925   invgcminusg 15926   -gcsg 15927   Abelcabl 16672
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  df-cmn 16673  df-abl 16674
This theorem is referenced by:  ngpinvds  21000  hdmap1neglem1N  37026
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