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Theorem ablsubsub4 16618
Description: Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
Hypotheses
Ref Expression
ablsubadd.b  |-  B  =  ( Base `  G
)
ablsubadd.p  |-  .+  =  ( +g  `  G )
ablsubadd.m  |-  .-  =  ( -g `  G )
ablsubsub.g  |-  ( ph  ->  G  e.  Abel )
ablsubsub.x  |-  ( ph  ->  X  e.  B )
ablsubsub.y  |-  ( ph  ->  Y  e.  B )
ablsubsub.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
ablsubsub4  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( X 
.-  ( Y  .+  Z ) ) )

Proof of Theorem ablsubsub4
StepHypRef Expression
1 ablsubsub.g . . . . 5  |-  ( ph  ->  G  e.  Abel )
2 ablgrp 16592 . . . . 5  |-  ( G  e.  Abel  ->  G  e. 
Grp )
31, 2syl 16 . . . 4  |-  ( ph  ->  G  e.  Grp )
4 ablsubsub.x . . . 4  |-  ( ph  ->  X  e.  B )
5 ablsubsub.y . . . 4  |-  ( ph  ->  Y  e.  B )
6 ablsubadd.b . . . . 5  |-  B  =  ( Base `  G
)
7 ablsubadd.m . . . . 5  |-  .-  =  ( -g `  G )
86, 7grpsubcl 15912 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  e.  B )
93, 4, 5, 8syl3anc 1223 . . 3  |-  ( ph  ->  ( X  .-  Y
)  e.  B )
10 ablsubsub.z . . 3  |-  ( ph  ->  Z  e.  B )
11 ablsubadd.p . . . 4  |-  .+  =  ( +g  `  G )
12 eqid 2460 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
136, 11, 12, 7grpsubval 15887 . . 3  |-  ( ( ( X  .-  Y
)  e.  B  /\  Z  e.  B )  ->  ( ( X  .-  Y )  .-  Z
)  =  ( ( X  .-  Y ) 
.+  ( ( invg `  G ) `
 Z ) ) )
149, 10, 13syl2anc 661 . 2  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( ( X  .-  Y ) 
.+  ( ( invg `  G ) `
 Z ) ) )
156, 12grpinvcl 15889 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( invg `  G ) `  Z
)  e.  B )
163, 10, 15syl2anc 661 . . 3  |-  ( ph  ->  ( ( invg `  G ) `  Z
)  e.  B )
176, 11, 7, 1, 4, 5, 16ablsubsub 16617 . 2  |-  ( ph  ->  ( X  .-  ( Y  .-  ( ( invg `  G ) `
 Z ) ) )  =  ( ( X  .-  Y ) 
.+  ( ( invg `  G ) `
 Z ) ) )
186, 11, 7, 12, 3, 5, 10grpsubinv 15905 . . 3  |-  ( ph  ->  ( Y  .-  (
( invg `  G ) `  Z
) )  =  ( Y  .+  Z ) )
1918oveq2d 6291 . 2  |-  ( ph  ->  ( X  .-  ( Y  .-  ( ( invg `  G ) `
 Z ) ) )  =  ( X 
.-  ( Y  .+  Z ) ) )
2014, 17, 193eqtr2d 2507 1  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( X 
.-  ( Y  .+  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716   invgcminusg 15717   -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:  ablsub32  16621  ip2subdi  18439  cpmadugsumlemF  19137  baerlem5alem2  36383
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