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Theorem ablinvadd 15111
Description: The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
ablinvadd.b  |-  B  =  ( Base `  G
)
ablinvadd.p  |-  .+  =  ( +g  `  G )
ablinvadd.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
ablinvadd  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `  X )  .+  ( N `  Y )
) )

Proof of Theorem ablinvadd
StepHypRef Expression
1 ablgrp 15094 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
2 ablinvadd.b . . . 4  |-  B  =  ( Base `  G
)
3 ablinvadd.p . . . 4  |-  .+  =  ( +g  `  G )
4 ablinvadd.n . . . 4  |-  N  =  ( inv g `  G )
52, 3, 4grpinvadd 14544 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `
 Y )  .+  ( N `  X ) ) )
61, 5syl3an1 1215 . 2  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `  Y )  .+  ( N `  X )
) )
7 simp1 955 . . 3  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  G  e.  Abel )
813ad2ant1 976 . . . 4  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  G  e.  Grp )
9 simp2 956 . . . 4  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
102, 4grpinvcl 14527 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
118, 9, 10syl2anc 642 . . 3  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  X )  e.  B )
12 simp3 957 . . . 4  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
132, 4grpinvcl 14527 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
148, 12, 13syl2anc 642 . . 3  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  Y )  e.  B )
152, 3ablcom 15106 . . 3  |-  ( ( G  e.  Abel  /\  ( N `  X )  e.  B  /\  ( N `  Y )  e.  B )  ->  (
( N `  X
)  .+  ( N `  Y ) )  =  ( ( N `  Y )  .+  ( N `  X )
) )
167, 11, 14, 15syl3anc 1182 . 2  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  (
( N `  X
)  .+  ( N `  Y ) )  =  ( ( N `  Y )  .+  ( N `  X )
) )
176, 16eqtr4d 2318 1  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `  X )  .+  ( N `  Y )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   inv gcminusg 14363   Abelcabel 15090
This theorem is referenced by:  ablsub4  15114  mulgdi  15126  invghm  15130  lmodnegadd  15674  lflnegcl  29265  baerlem3lem1  31897
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
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-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-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-cmn 15091  df-abl 15092
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