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Theorem grpaddsubass 15595
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 454 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  G  e.  Grp )
2 simpr1 987 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
3 simpr2 988 . . 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 2433 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
64, 5grpinvcl 15563 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( invg `  G ) `  Z
)  e.  B )
763ad2antr3 1148 . . 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 15532 . . 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 1213 . 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 15531 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y
)  e.  B )
12113adant3r3 1191 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  Y )  e.  B )
13 simpr3 989 . . 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 15561 . . 3  |-  ( ( ( X  .+  Y
)  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Y )  .-  Z
)  =  ( ( X  .+  Y ) 
.+  ( ( invg `  G ) `
 Z ) ) )
1612, 13, 15syl2anc 654 . 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 15561 . . . 4  |-  ( ( Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( invg `  G ) `
 Z ) ) )
183, 13, 17syl2anc 654 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .-  Z )  =  ( Y  .+  (
( invg `  G ) `  Z
) ) )
1918oveq2d 6096 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  ( Y  .-  Z ) )  =  ( X  .+  ( Y  .+  ( ( invg `  G ) `
 Z ) ) ) )
2010, 16, 193eqtr4d 2475 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 958    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Basecbs 14157   +g cplusg 14221   Grpcgrp 15393   invgcminusg 15394   -gcsg 15396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-0g 14363  df-mnd 15398  df-grp 15525  df-minusg 15526  df-sbg 15527
This theorem is referenced by:  grppncan  15596  grpnpncan  15600  nsgconj  15694  conjghm  15757  conjnmz  15760  conjnmzb  15761  sylow3lem1  16106  sylow3lem2  16107  abladdsub  16284  ablsubsub  16287  archiabllem2a  26035
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