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Theorem abladdsub 16614
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 16604 . . . 4  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
433adant3r3 1202 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
54oveq1d 6290 . 2  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .-  Z )  =  ( ( Y  .+  X
)  .-  Z )
)
6 ablgrp 16592 . . . 4  |-  ( G  e.  Abel  ->  G  e. 
Grp )
76adantr 465 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  G  e.  Grp )
8 simpr2 998 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Y  e.  B )
9 simpr1 997 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  X  e.  B )
10 simpr3 999 . . 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 15922 . . 3  |-  ( ( G  e.  Grp  /\  ( Y  e.  B  /\  X  e.  B  /\  Z  e.  B
) )  ->  (
( Y  .+  X
)  .-  Z )  =  ( Y  .+  ( X  .-  Z ) ) )
137, 8, 9, 10, 12syl13anc 1225 . 2  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( Y  .+  X )  .-  Z )  =  ( Y  .+  ( X 
.-  Z ) ) )
14 simpl 457 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  G  e.  Abel )
151, 11grpsubcl 15912 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .-  Z
)  e.  B )
167, 9, 10, 15syl3anc 1223 . . 3  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .-  Z )  e.  B
)
171, 2ablcom 16604 . . 3  |-  ( ( G  e.  Abel  /\  Y  e.  B  /\  ( X  .-  Z )  e.  B )  ->  ( Y  .+  ( X  .-  Z ) )  =  ( ( X  .-  Z )  .+  Y
) )
1814, 8, 16, 17syl3anc 1223 . 2  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .+  ( X  .-  Z
) )  =  ( ( X  .-  Z
)  .+  Y )
)
195, 13, 183eqtrd 2505 1  |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .-  Z )  =  ( ( X  .-  Z
)  .+  Y )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716   -gcsg 15719   Abelcabel 16588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-cmn 16589  df-abl 16590
This theorem is referenced by:  ablpncan2  16615  ablsubsub  16617  ip2subdi  18439
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