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Theorem ablinvadd 15434
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 15417 . . 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 14867 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `
 Y )  .+  ( N `  X ) ) )
61, 5syl3an1 1217 . 2  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `  Y )  .+  ( N `  X )
) )
7 simp1 957 . . 3  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  G  e.  Abel )
813ad2ant1 978 . . . 4  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  G  e.  Grp )
9 simp2 958 . . . 4  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
102, 4grpinvcl 14850 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
118, 9, 10syl2anc 643 . . 3  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  X )  e.  B )
12 simp3 959 . . . 4  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
132, 4grpinvcl 14850 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
148, 12, 13syl2anc 643 . . 3  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  Y )  e.  B )
152, 3ablcom 15429 . . 3  |-  ( ( G  e.  Abel  /\  ( N `  X )  e.  B  /\  ( N `  Y )  e.  B )  ->  (
( N `  X
)  .+  ( N `  Y ) )  =  ( ( N `  Y )  .+  ( N `  X )
) )
167, 11, 14, 15syl3anc 1184 . 2  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  (
( N `  X
)  .+  ( N `  Y ) )  =  ( ( N `  Y )  .+  ( N `  X )
) )
176, 16eqtr4d 2471 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 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685   inv gcminusg 14686   Abelcabel 15413
This theorem is referenced by:  ablsub4  15437  mulgdi  15449  invghm  15453  lmodnegadd  15993  lflnegcl  29873  baerlem3lem1  32505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-cmn 15414  df-abl 15415
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