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Theorem ablsubsub4 15445
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 15419 . . . . 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 14871 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  e.  B )
93, 4, 5, 8syl3anc 1185 . . 3  |-  ( ph  ->  ( X  .-  Y
)  e.  B )
10 ablsubsub.z . . 3  |-  ( ph  ->  Z  e.  B )
11 ablsubadd.p . . . 4  |-  .+  =  ( +g  `  G )
12 eqid 2438 . . . 4  |-  ( inv g `  G )  =  ( inv g `  G )
136, 11, 12, 7grpsubval 14850 . . 3  |-  ( ( ( X  .-  Y
)  e.  B  /\  Z  e.  B )  ->  ( ( X  .-  Y )  .-  Z
)  =  ( ( X  .-  Y ) 
.+  ( ( inv g `  G ) `
 Z ) ) )
149, 10, 13syl2anc 644 . 2  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( ( X  .-  Y ) 
.+  ( ( inv g `  G ) `
 Z ) ) )
156, 12grpinvcl 14852 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( inv g `  G ) `  Z
)  e.  B )
163, 10, 15syl2anc 644 . . 3  |-  ( ph  ->  ( ( inv g `  G ) `  Z
)  e.  B )
176, 11, 7, 1, 4, 5, 16ablsubsub 15444 . 2  |-  ( ph  ->  ( X  .-  ( Y  .-  ( ( inv g `  G ) `
 Z ) ) )  =  ( ( X  .-  Y ) 
.+  ( ( inv g `  G ) `
 Z ) ) )
186, 11, 7, 12, 3, 5, 10grpsubinv 14866 . . 3  |-  ( ph  ->  ( Y  .-  (
( inv g `  G ) `  Z
) )  =  ( Y  .+  Z ) )
1918oveq2d 6099 . 2  |-  ( ph  ->  ( X  .-  ( Y  .-  ( ( inv g `  G ) `
 Z ) ) )  =  ( X 
.-  ( Y  .+  Z ) ) )
2014, 17, 193eqtr2d 2476 1  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( X 
.-  ( Y  .+  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   Grpcgrp 14687   inv gcminusg 14688   -gcsg 14690   Abelcabel 15415
This theorem is referenced by:  ablsub32  15448  ip2subdi  16877  baerlem5alem2  32571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-cmn 15416  df-abl 15417
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