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Theorem cmn4 15433
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 958 . . 3  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  G  e. CMnd )
4 cmnmnd 15429 . . 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 984 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  X  e.  B )
7 simp2r 985 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  Y  e.  B )
8 simp3l 986 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  Z  e.  B )
9 simp3r 987 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  W  e.  B )
101, 2cmncom 15430 . . 3  |-  ( ( G  e. CMnd  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
113, 7, 8, 10syl3anc 1185 . 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 14703 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   Mndcmnd 14686  CMndccmn 15414
This theorem is referenced by:  ablsub4  15439  ghmplusg  15463  lmod4  15996  ip2di  16874  evlslem1  19938  lfladdcl  29931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340  ax-pow 4379
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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-mnd 14692  df-cmn 15416
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