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Theorem grpaddsubass 15997
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 1001 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
3 simpr2 1002 . . 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 2441 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
64, 5grpinvcl 15964 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( invg `  G ) `  Z
)  e.  B )
763ad2antr3 1162 . . 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 15933 . . 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 1229 . 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 15932 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y
)  e.  B )
12113adant3r3 1206 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  Y )  e.  B )
13 simpr3 1003 . . 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 15962 . . 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 15962 . . . 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 2492 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 972    = wceq 1381    e. wcel 1802   ` cfv 5574  (class class class)co 6277   Basecbs 14504   +g cplusg 14569   Grpcgrp 15922   invgcminusg 15923   -gcsg 15924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-sbg 15928
This theorem is referenced by:  grppncan  15998  grpnpncan  16002  nsgconj  16103  conjghm  16166  conjnmz  16169  conjnmzb  16170  sylow3lem1  16516  sylow3lem2  16517  abladdsub  16694  ablsubsub  16697  cpmadugsumlemF  19244  archiabllem2a  27604
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