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Theorem abl32 15110
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 15095 . . 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 15107 . 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  CMndccmn 15089   Abelcabel 15090
This theorem is referenced by:  baerlem5alem1  31898  baerlem5blem1  31899
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  df-abl 15092
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