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Theorem ablsub32 15366
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 2380 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
64, 5ablcom 15349 . . . 4  |-  ( ( G  e.  Abel  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y ( +g  `  G
) Z )  =  ( Z ( +g  `  G ) Y ) )
71, 2, 3, 6syl3anc 1184 . . 3  |-  ( ph  ->  ( Y ( +g  `  G ) Z )  =  ( Z ( +g  `  G ) Y ) )
87oveq2d 6029 . 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 15363 . 2  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( X 
.-  ( Y ( +g  `  G ) Z ) ) )
124, 5, 9, 1, 10, 3, 2ablsubsub4 15363 . 2  |-  ( ph  ->  ( ( X  .-  Z )  .-  Y
)  =  ( X 
.-  ( Z ( +g  `  G ) Y ) ) )
138, 11, 123eqtr4d 2422 1  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( ( X  .-  Z ) 
.-  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   Basecbs 13389   +g cplusg 13449   -gcsg 14608   Abelcabel 15333
This theorem is referenced by:  ablnnncan1  15367  baerlem5alem2  31877
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-0g 13647  df-mnd 14610  df-grp 14732  df-minusg 14733  df-sbg 14734  df-cmn 15334  df-abl 15335
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