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Theorem ablsub2inv 15437
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  =  ( inv g `  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 2438 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 ablsub2inv.m . . 3  |-  .-  =  ( -g `  G )
4 ablsub2inv.n . . 3  |-  N  =  ( inv g `  G )
5 ablsub2inv.g . . . 4  |-  ( ph  ->  G  e.  Abel )
6 ablgrp 15419 . . . 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 14852 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
107, 8, 9syl2anc 644 . . 3  |-  ( ph  ->  ( N `  X
)  e.  B )
11 ablsub2inv.y . . 3  |-  ( ph  ->  Y  e.  B )
121, 2, 3, 4, 7, 10, 11grpsubinv 14866 . 2  |-  ( ph  ->  ( ( N `  X )  .-  ( N `  Y )
)  =  ( ( N `  X ) ( +g  `  G
) Y ) )
131, 2ablcom 15431 . . . . . 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 1185 . . . . 5  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( Y ( +g  `  G ) ( N `  X
) ) )
151, 4grpinvinv 14860 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
167, 11, 15syl2anc 644 . . . . . 6  |-  ( ph  ->  ( N `  ( N `  Y )
)  =  Y )
1716oveq1d 6098 . . . . 5  |-  ( ph  ->  ( ( N `  ( N `  Y ) ) ( +g  `  G
) ( N `  X ) )  =  ( Y ( +g  `  G ) ( N `
 X ) ) )
1814, 17eqtr4d 2473 . . . 4  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
191, 4grpinvcl 14852 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
207, 11, 19syl2anc 644 . . . . 5  |-  ( ph  ->  ( N `  Y
)  e.  B )
211, 2, 4grpinvadd 14869 . . . . 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 1185 . . . 4  |-  ( ph  ->  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
2318, 22eqtr4d 2473 . . 3  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
241, 2, 4, 3grpsubval 14850 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
258, 11, 24syl2anc 644 . . . 4  |-  ( ph  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
2625fveq2d 5734 . . 3  |-  ( ph  ->  ( N `  ( X  .-  Y ) )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
2723, 26eqtr4d 2473 . 2  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( N `  ( X  .-  Y ) ) )
281, 3, 4grpinvsub 14873 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
297, 8, 11, 28syl3anc 1185 . 2  |-  ( ph  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
3012, 27, 293eqtrd 2474 1  |-  ( ph  ->  ( ( N `  X )  .-  ( N `  Y )
)  =  ( Y 
.-  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   Grpcgrp 14687   inv gcminusg 14688   -gcsg 14690   Abelcabel 15415
This theorem is referenced by:  ngpinvds  18661  hdmap1neglem1N  32688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-cmn 15416  df-abl 15417
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