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Theorem abl32 15126
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 15111 . . 3  |-  ( G  e.  Abel  ->  G  e. CMnd
)
31, 2syl 15 . 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 15123 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .+  Z )  =  ( ( X  .+  Z
)  .+  Y )
)
103, 4, 5, 6, 9syl13anc 1184 1  |-  ( ph  ->  ( ( X  .+  Y )  .+  Z
)  =  ( ( X  .+  Z ) 
.+  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  CMndccmn 15105   Abelcabel 15106
This theorem is referenced by:  baerlem5alem1  32520  baerlem5blem1  32521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-mnd 14383  df-cmn 15107  df-abl 15108
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