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Theorem ablsub2inv 16300
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 2443 . . 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 16282 . . . 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 15583 . . . 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 15599 . 2  |-  ( ph  ->  ( ( N `  X )  .-  ( N `  Y )
)  =  ( ( N `  X ) ( +g  `  G
) Y ) )
131, 2ablcom 16294 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( Y ( +g  `  G ) ( N `  X
) ) )
151, 4grpinvinv 15593 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
167, 11, 15syl2anc 661 . . . . . 6  |-  ( ph  ->  ( N `  ( N `  Y )
)  =  Y )
1716oveq1d 6106 . . . . 5  |-  ( ph  ->  ( ( N `  ( N `  Y ) ) ( +g  `  G
) ( N `  X ) )  =  ( Y ( +g  `  G ) ( N `
 X ) ) )
1814, 17eqtr4d 2478 . . . 4  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
191, 4grpinvcl 15583 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
207, 11, 19syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  Y
)  e.  B )
211, 2, 4grpinvadd 15604 . . . . 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 1218 . . . 4  |-  ( ph  ->  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
2318, 22eqtr4d 2478 . . 3  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
241, 2, 4, 3grpsubval 15581 . . . . 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 5695 . . 3  |-  ( ph  ->  ( N `  ( X  .-  Y ) )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
2723, 26eqtr4d 2478 . 2  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( N `  ( X  .-  Y ) ) )
281, 3, 4grpinvsub 15608 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
297, 8, 11, 28syl3anc 1218 . 2  |-  ( ph  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
3012, 27, 293eqtrd 2479 1  |-  ( ph  ->  ( ( N `  X )  .-  ( N `  Y )
)  =  ( Y 
.-  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   Grpcgrp 15410   invgcminusg 15411   -gcsg 15413   Abelcabel 16278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-cmn 16279  df-abl 16280
This theorem is referenced by:  ngpinvds  20204  hdmap1neglem1N  35473
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