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Theorem abladdsub 15439
Description: Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
Hypotheses
Ref Expression
ablsubadd.b  |-  B  =  ( Base `  G
)
ablsubadd.p  |-  .+  =  ( +g  `  G )
ablsubadd.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
abladdsub  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .-  Z )  =  ( ( X  .-  Z
)  .+  Y )
)

Proof of Theorem abladdsub
StepHypRef Expression
1 ablsubadd.b . . . . 5  |-  B  =  ( Base `  G
)
2 ablsubadd.p . . . . 5  |-  .+  =  ( +g  `  G )
31, 2ablcom 15429 . . . 4  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
433adant3r3 1164 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
54oveq1d 6096 . 2  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .-  Z )  =  ( ( Y  .+  X
)  .-  Z )
)
6 ablgrp 15417 . . . 4  |-  ( G  e.  Abel  ->  G  e. 
Grp )
76adantr 452 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  G  e.  Grp )
8 simpr2 964 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Y  e.  B )
9 simpr1 963 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  X  e.  B )
10 simpr3 965 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  e.  B )
11 ablsubadd.m . . . 4  |-  .-  =  ( -g `  G )
121, 2, 11grpaddsubass 14878 . . 3  |-  ( ( G  e.  Grp  /\  ( Y  e.  B  /\  X  e.  B  /\  Z  e.  B
) )  ->  (
( Y  .+  X
)  .-  Z )  =  ( Y  .+  ( X  .-  Z ) ) )
137, 8, 9, 10, 12syl13anc 1186 . 2  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( Y  .+  X )  .-  Z )  =  ( Y  .+  ( X 
.-  Z ) ) )
14 simpl 444 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  G  e.  Abel )
151, 11grpsubcl 14869 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .-  Z
)  e.  B )
167, 9, 10, 15syl3anc 1184 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .-  Z )  e.  B
)
171, 2ablcom 15429 . . 3  |-  ( ( G  e.  Abel  /\  Y  e.  B  /\  ( X  .-  Z )  e.  B )  ->  ( Y  .+  ( X  .-  Z ) )  =  ( ( X  .-  Z )  .+  Y
) )
1814, 8, 16, 17syl3anc 1184 . 2  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .+  ( X  .-  Z
) )  =  ( ( X  .-  Z
)  .+  Y )
)
195, 13, 183eqtrd 2472 1  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .-  Z )  =  ( ( X  .-  Z
)  .+  Y )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685   -gcsg 14688   Abelcabel 15413
This theorem is referenced by:  ablpncan2  15440  ablsubsub  15442  ip2subdi  16875
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  ax-un 4701
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-pw 3801  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-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-cmn 15414  df-abl 15415
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