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Theorem ablsub32 15484
Description: Swap the second and third terms in a double subtraction. (Contributed by NM, 7-Apr-2015.)
Hypotheses
Ref Expression
ablnncan.b  |-  B  =  ( Base `  G
)
ablnncan.m  |-  .-  =  ( -g `  G )
ablnncan.g  |-  ( ph  ->  G  e.  Abel )
ablnncan.x  |-  ( ph  ->  X  e.  B )
ablnncan.y  |-  ( ph  ->  Y  e.  B )
ablsub32.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
ablsub32  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( ( X  .-  Z ) 
.-  Y ) )

Proof of Theorem ablsub32
StepHypRef Expression
1 ablnncan.g . . . 4  |-  ( ph  ->  G  e.  Abel )
2 ablnncan.y . . . 4  |-  ( ph  ->  Y  e.  B )
3 ablsub32.z . . . 4  |-  ( ph  ->  Z  e.  B )
4 ablnncan.b . . . . 5  |-  B  =  ( Base `  G
)
5 eqid 2443 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
64, 5ablcom 15467 . . . 4  |-  ( ( G  e.  Abel  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y ( +g  `  G
) Z )  =  ( Z ( +g  `  G ) Y ) )
71, 2, 3, 6syl3anc 1185 . . 3  |-  ( ph  ->  ( Y ( +g  `  G ) Z )  =  ( Z ( +g  `  G ) Y ) )
87oveq2d 6133 . 2  |-  ( ph  ->  ( X  .-  ( Y ( +g  `  G
) Z ) )  =  ( X  .-  ( Z ( +g  `  G
) Y ) ) )
9 ablnncan.m . . 3  |-  .-  =  ( -g `  G )
10 ablnncan.x . . 3  |-  ( ph  ->  X  e.  B )
114, 5, 9, 1, 10, 2, 3ablsubsub4 15481 . 2  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( X 
.-  ( Y ( +g  `  G ) Z ) ) )
124, 5, 9, 1, 10, 3, 2ablsubsub4 15481 . 2  |-  ( ph  ->  ( ( X  .-  Z )  .-  Y
)  =  ( X 
.-  ( Z ( +g  `  G ) Y ) ) )
138, 11, 123eqtr4d 2485 1  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( ( X  .-  Z ) 
.-  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1654    e. wcel 1728   ` cfv 5489  (class class class)co 6117   Basecbs 13507   +g cplusg 13567   -gcsg 14726   Abelcabel 15451
This theorem is referenced by:  ablnnncan1  15485  baerlem5alem2  32683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-riota 6585  df-0g 13765  df-mnd 14728  df-grp 14850  df-minusg 14851  df-sbg 14852  df-cmn 15452  df-abl 15453
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