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Theorem cmn32 15358
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
cmn32  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .+  Z )  =  ( ( X  .+  Z
)  .+  Y )
)

Proof of Theorem cmn32
StepHypRef Expression
1 ablcom.b . 2  |-  B  =  ( Base `  G
)
2 ablcom.p . 2  |-  .+  =  ( +g  `  G )
3 cmnmnd 15355 . . 3  |-  ( G  e. CMnd  ->  G  e.  Mnd )
43adantr 452 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  G  e.  Mnd )
5 simpr1 963 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  X  e.  B )
6 simpr2 964 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Y  e.  B )
7 simpr3 965 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  e.  B )
81, 2cmncom 15356 . . 3  |-  ( ( G  e. CMnd  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
983adant3r1 1162 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .+  Z )  =  ( Z  .+  Y ) )
101, 2, 4, 5, 6, 7, 9mnd32g 14627 1  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .+  Z )  =  ( ( X  .+  Z
)  .+  Y )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   Mndcmnd 14612  CMndccmn 15340
This theorem is referenced by:  abl32  15361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-nul 4280  ax-pow 4319
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024  df-mnd 14618  df-cmn 15342
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