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Theorem ablo32 21866
Description: Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
ablcom.1  |-  X  =  ran  G
Assertion
Ref Expression
ablo32  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )

Proof of Theorem ablo32
StepHypRef Expression
1 ablcom.1 . . . . 5  |-  X  =  ran  G
21ablocom 21865 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B G C )  =  ( C G B ) )
323adant3r1 1162 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B G C )  =  ( C G B ) )
43oveq2d 6089 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( B G C ) )  =  ( A G ( C G B ) ) )
5 ablogrpo 21864 . . 3  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
61grpoass 21783 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
75, 6sylan 458 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
8 3ancomb 945 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)
91grpoass 21783 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A G C ) G B )  =  ( A G ( C G B ) ) )
108, 9sylan2b 462 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G C ) G B )  =  ( A G ( C G B ) ) )
115, 10sylan 458 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G C ) G B )  =  ( A G ( C G B ) ) )
124, 7, 113eqtr4d 2477 1  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ran crn 4871  (class class class)co 6073   GrpOpcgr 21766   AbelOpcablo 21861
This theorem is referenced by:  ablo4  21867  rngoa32  21974  vca32  22033  nvadd32  22095  ip0i  22318
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-grpo 21771  df-ablo 21862
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