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Theorem cmn4 16296
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 988 . . 3  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  G  e. CMnd )
4 cmnmnd 16292 . . 3  |-  ( G  e. CMnd  ->  G  e.  Mnd )
53, 4syl 16 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  G  e.  Mnd )
6 simp2l 1014 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  X  e.  B )
7 simp2r 1015 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  Y  e.  B )
8 simp3l 1016 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  Z  e.  B )
9 simp3r 1017 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  W  e.  B )
101, 2cmncom 16293 . . 3  |-  ( ( G  e. CMnd  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
113, 7, 8, 10syl3anc 1218 . 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 15426 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   Mndcmnd 15409  CMndccmn 16277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4421  ax-pow 4470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-mnd 15415  df-cmn 16279
This theorem is referenced by:  ablsub4  16302  ghmplusg  16328  lmod4  16995  evlslem1  17601  ip2di  18070  lfladdcl  32716
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