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Theorem cmn4 17384
Description: Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
ablcom.b  |-  B  =  ( Base `  G
)
ablcom.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
cmn4  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  (
( X  .+  Y
)  .+  ( Z  .+  W ) )  =  ( ( X  .+  Z )  .+  ( Y  .+  W ) ) )

Proof of Theorem cmn4
StepHypRef Expression
1 ablcom.b . 2  |-  B  =  ( Base `  G
)
2 ablcom.p . 2  |-  .+  =  ( +g  `  G )
3 simp1 1005 . . 3  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  G  e. CMnd )
4 cmnmnd 17380 . . 3  |-  ( G  e. CMnd  ->  G  e.  Mnd )
53, 4syl 17 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  G  e.  Mnd )
6 simp2l 1031 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  X  e.  B )
7 simp2r 1032 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  Y  e.  B )
8 simp3l 1033 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  Z  e.  B )
9 simp3r 1034 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  W  e.  B )
101, 2cmncom 17381 . . 3  |-  ( ( G  e. CMnd  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
113, 7, 8, 10syl3anc 1264 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
121, 2, 5, 6, 7, 8, 9, 11mnd4g 16504 1  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  (
( X  .+  Y
)  .+  ( Z  .+  W ) )  =  ( ( X  .+  Z )  .+  ( Y  .+  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   Mndcmnd 16486  CMndccmn 17365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-nul 4556
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609  df-ov 6308  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-cmn 17367
This theorem is referenced by:  ablsub4  17390  ghmplusg  17419  lmod4  18073  evlslem1  18673  ip2di  19139  lfladdcl  32346
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