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Theorem ablsub32 15429
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 2430 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
64, 5ablcom 15412 . . . 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 6083 . 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 15426 . 2  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( X 
.-  ( Y ( +g  `  G ) Z ) ) )
124, 5, 9, 1, 10, 3, 2ablsubsub4 15426 . 2  |-  ( ph  ->  ( ( X  .-  Z )  .-  Y
)  =  ( X 
.-  ( Z ( +g  `  G ) Y ) ) )
138, 11, 123eqtr4d 2472 1  |-  ( ph  ->  ( ( X  .-  Y )  .-  Z
)  =  ( ( X  .-  Z ) 
.-  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5440  (class class class)co 6067   Basecbs 13452   +g cplusg 13512   -gcsg 14671   Abelcabel 15396
This theorem is referenced by:  ablnnncan1  15430  baerlem5alem2  32240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-0g 13710  df-mnd 14673  df-grp 14795  df-minusg 14796  df-sbg 14797  df-cmn 15397  df-abl 15398
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