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Theorem grpaddsubass 15924
Description: Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
grpsubadd.b  |-  B  =  ( Base `  G
)
grpsubadd.p  |-  .+  =  ( +g  `  G )
grpsubadd.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grpaddsubass  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .-  Z )  =  ( X  .+  ( Y  .-  Z ) ) )

Proof of Theorem grpaddsubass
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  G  e.  Grp )
2 simpr1 997 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
3 simpr2 998 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
4 grpsubadd.b . . . . 5  |-  B  =  ( Base `  G
)
5 eqid 2462 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
64, 5grpinvcl 15891 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( invg `  G ) `  Z
)  e.  B )
763ad2antr3 1158 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( invg `  G ) `  Z
)  e.  B )
8 grpsubadd.p . . . 4  |-  .+  =  ( +g  `  G )
94, 8grpass 15860 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  ( ( invg `  G ) `  Z
)  e.  B ) )  ->  ( ( X  .+  Y )  .+  ( ( invg `  G ) `  Z
) )  =  ( X  .+  ( Y 
.+  ( ( invg `  G ) `
 Z ) ) ) )
101, 2, 3, 7, 9syl13anc 1225 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .+  ( ( invg `  G ) `
 Z ) )  =  ( X  .+  ( Y  .+  ( ( invg `  G
) `  Z )
) ) )
114, 8grpcl 15859 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y
)  e.  B )
12113adant3r3 1202 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  Y )  e.  B )
13 simpr3 999 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
14 grpsubadd.m . . . 4  |-  .-  =  ( -g `  G )
154, 8, 5, 14grpsubval 15889 . . 3  |-  ( ( ( X  .+  Y
)  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Y )  .-  Z
)  =  ( ( X  .+  Y ) 
.+  ( ( invg `  G ) `
 Z ) ) )
1612, 13, 15syl2anc 661 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .-  Z )  =  ( ( X 
.+  Y )  .+  ( ( invg `  G ) `  Z
) ) )
174, 8, 5, 14grpsubval 15889 . . . 4  |-  ( ( Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( invg `  G ) `
 Z ) ) )
183, 13, 17syl2anc 661 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .-  Z )  =  ( Y  .+  (
( invg `  G ) `  Z
) ) )
1918oveq2d 6293 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  ( Y  .-  Z ) )  =  ( X  .+  ( Y  .+  ( ( invg `  G ) `
 Z ) ) ) )
2010, 16, 193eqtr4d 2513 1  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .-  Z )  =  ( X  .+  ( Y  .-  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546   Grpcgrp 15718   invgcminusg 15719   -gcsg 15721
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-0g 14688  df-mnd 15723  df-grp 15853  df-minusg 15854  df-sbg 15855
This theorem is referenced by:  grppncan  15925  grpnpncan  15929  nsgconj  16024  conjghm  16087  conjnmz  16090  conjnmzb  16091  sylow3lem1  16438  sylow3lem2  16439  abladdsub  16616  ablsubsub  16619  cpmadugsumlemF  19139  archiabllem2a  27388
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