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Theorem abl32 15433
Description: Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
ablcom.b  |-  B  =  ( Base `  G
)
ablcom.p  |-  .+  =  ( +g  `  G )
abl32.g  |-  ( ph  ->  G  e.  Abel )
abl32.x  |-  ( ph  ->  X  e.  B )
abl32.y  |-  ( ph  ->  Y  e.  B )
abl32.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
abl32  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( ( X  .+  Z ) 
.+  Y ) )

Proof of Theorem abl32
StepHypRef Expression
1 abl32.g . . 3  |-  ( ph  ->  G  e.  Abel )
2 ablcmn 15418 . . 3  |-  ( G  e.  Abel  ->  G  e. CMnd
)
31, 2syl 16 . 2  |-  ( ph  ->  G  e. CMnd )
4 abl32.x . 2  |-  ( ph  ->  X  e.  B )
5 abl32.y . 2  |-  ( ph  ->  Y  e.  B )
6 abl32.z . 2  |-  ( ph  ->  Z  e.  B )
7 ablcom.b . . 3  |-  B  =  ( Base `  G
)
8 ablcom.p . . 3  |-  .+  =  ( +g  `  G )
97, 8cmn32 15430 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .+  Z )  =  ( ( X  .+  Z
)  .+  Y )
)
103, 4, 5, 6, 9syl13anc 1186 1  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( ( X  .+  Z ) 
.+  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529  CMndccmn 15412   Abelcabel 15413
This theorem is referenced by:  baerlem5alem1  32506  baerlem5blem1  32507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338  ax-pow 4377
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-mnd 14690  df-cmn 15414  df-abl 15415
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