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Theorem cmn4 15108
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 955 . . 3  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  G  e. CMnd )
4 cmnmnd 15104 . . 3  |-  ( G  e. CMnd  ->  G  e.  Mnd )
53, 4syl 15 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  G  e.  Mnd )
6 simp2l 981 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  X  e.  B )
7 simp2r 982 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  Y  e.  B )
8 simp3l 983 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  Z  e.  B )
9 simp3r 984 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B
) )  ->  W  e.  B )
101, 2cmncom 15105 . . 3  |-  ( ( G  e. CMnd  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
113, 7, 8, 10syl3anc 1182 . 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 14378 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Mndcmnd 14361  CMndccmn 15089
This theorem is referenced by:  ablsub4  15114  ghmplusg  15138  lmod4  15675  ip2di  16545  evlslem1  19399  lfladdcl  29261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-mnd 14367  df-cmn 15091
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