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Theorem grpaddsubass 14833
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 444 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  G  e.  Grp )
2 simpr1 963 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
3 simpr2 964 . . 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 2404 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
64, 5grpinvcl 14805 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( inv g `  G ) `  Z
)  e.  B )
763ad2antr3 1124 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( inv g `  G ) `  Z
)  e.  B )
8 grpsubadd.p . . . 4  |-  .+  =  ( +g  `  G )
94, 8grpass 14774 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  ( ( inv g `  G ) `  Z
)  e.  B ) )  ->  ( ( X  .+  Y )  .+  ( ( inv g `  G ) `  Z
) )  =  ( X  .+  ( Y 
.+  ( ( inv g `  G ) `
 Z ) ) ) )
101, 2, 3, 7, 9syl13anc 1186 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .+  ( ( inv g `  G ) `
 Z ) )  =  ( X  .+  ( Y  .+  ( ( inv g `  G
) `  Z )
) ) )
114, 8grpcl 14773 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y
)  e.  B )
12113adant3r3 1164 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  Y )  e.  B )
13 simpr3 965 . . 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 14803 . . 3  |-  ( ( ( X  .+  Y
)  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Y )  .-  Z
)  =  ( ( X  .+  Y ) 
.+  ( ( inv g `  G ) `
 Z ) ) )
1612, 13, 15syl2anc 643 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .-  Z )  =  ( ( X 
.+  Y )  .+  ( ( inv g `  G ) `  Z
) ) )
174, 8, 5, 14grpsubval 14803 . . . 4  |-  ( ( Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  G ) `
 Z ) ) )
183, 13, 17syl2anc 643 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .-  Z )  =  ( Y  .+  (
( inv g `  G ) `  Z
) ) )
1918oveq2d 6056 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  ( Y  .-  Z ) )  =  ( X  .+  ( Y  .+  ( ( inv g `  G ) `
 Z ) ) ) )
2010, 16, 193eqtr4d 2446 1  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Y
)  .-  Z )  =  ( X  .+  ( Y  .-  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   Grpcgrp 14640   inv gcminusg 14641   -gcsg 14643
This theorem is referenced by:  grppncan  14834  grpnpncan  14838  nsgconj  14928  conjghm  14991  conjnmz  14994  conjnmzb  14995  sylow3lem1  15216  sylow3lem2  15217  abladdsub  15394  ablsubsub  15397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769
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