MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablo32 Structured version   Unicode version

Theorem ablo32 23785
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 23784 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B G C )  =  ( C G B ) )
323adant3r1 1196 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B G C )  =  ( C G B ) )
43oveq2d 6119 . 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 23783 . . 3  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
61grpoass 23702 . . 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 471 . 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 974 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)
91grpoass 23702 . . . 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 475 . . 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 471 . 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 2485 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ran crn 4853  (class class class)co 6103   GrpOpcgr 23685   AbelOpcablo 23780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fo 5436  df-fv 5438  df-ov 6106  df-grpo 23690  df-ablo 23781
This theorem is referenced by:  ablo4  23786  rngoa32  23893  vca32  23952  nvadd32  24014  ip0i  24237
  Copyright terms: Public domain W3C validator