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Theorem ablsub2inv 15112
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 2283 . . 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 15094 . . . 4  |-  ( G  e.  Abel  ->  G  e. 
Grp )
75, 6syl 15 . . 3  |-  ( ph  ->  G  e.  Grp )
8 ablsub2inv.x . . . 4  |-  ( ph  ->  X  e.  B )
91, 4grpinvcl 14527 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
107, 8, 9syl2anc 642 . . 3  |-  ( ph  ->  ( N `  X
)  e.  B )
11 ablsub2inv.y . . 3  |-  ( ph  ->  Y  e.  B )
121, 2, 3, 4, 7, 10, 11grpsubinv 14541 . 2  |-  ( ph  ->  ( ( N `  X )  .-  ( N `  Y )
)  =  ( ( N `  X ) ( +g  `  G
) Y ) )
131, 2ablcom 15106 . . . . . 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 1182 . . . . 5  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( Y ( +g  `  G ) ( N `  X
) ) )
151, 4grpinvinv 14535 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
167, 11, 15syl2anc 642 . . . . . 6  |-  ( ph  ->  ( N `  ( N `  Y )
)  =  Y )
1716oveq1d 5873 . . . . 5  |-  ( ph  ->  ( ( N `  ( N `  Y ) ) ( +g  `  G
) ( N `  X ) )  =  ( Y ( +g  `  G ) ( N `
 X ) ) )
1814, 17eqtr4d 2318 . . . 4  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
191, 4grpinvcl 14527 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
207, 11, 19syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  Y
)  e.  B )
211, 2, 4grpinvadd 14544 . . . . 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 1182 . . . 4  |-  ( ph  ->  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) )  =  ( ( N `
 ( N `  Y ) ) ( +g  `  G ) ( N `  X
) ) )
2318, 22eqtr4d 2318 . . 3  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
241, 2, 4, 3grpsubval 14525 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
258, 11, 24syl2anc 642 . . . 4  |-  ( ph  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( N `  Y ) ) )
2625fveq2d 5529 . . 3  |-  ( ph  ->  ( N `  ( X  .-  Y ) )  =  ( N `  ( X ( +g  `  G
) ( N `  Y ) ) ) )
2723, 26eqtr4d 2318 . 2  |-  ( ph  ->  ( ( N `  X ) ( +g  `  G ) Y )  =  ( N `  ( X  .-  Y ) ) )
281, 3, 4grpinvsub 14548 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
297, 8, 11, 28syl3anc 1182 . 2  |-  ( ph  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
3012, 27, 293eqtrd 2319 1  |-  ( ph  ->  ( ( N `  X )  .-  ( N `  Y )
)  =  ( Y 
.-  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365   Abelcabel 15090
This theorem is referenced by:  ngpinvds  18134  hdmap1neglem1N  32018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-cmn 15091  df-abl 15092
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