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Theorem ablocom 23772
Description: An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
ablcom.1  |-  X  =  ran  G
Assertion
Ref Expression
ablocom  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )

Proof of Theorem ablocom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablcom.1 . . . . 5  |-  X  =  ran  G
21isablo 23770 . . . 4  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
32simprbi 464 . . 3  |-  ( G  e.  AbelOp  ->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) )
4 oveq1 6098 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
5 oveq2 6099 . . . . 5  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
64, 5eqeq12d 2457 . . . 4  |-  ( x  =  A  ->  (
( x G y )  =  ( y G x )  <->  ( A G y )  =  ( y G A ) ) )
7 oveq2 6099 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
8 oveq1 6098 . . . . 5  |-  ( y  =  B  ->  (
y G A )  =  ( B G A ) )
97, 8eqeq12d 2457 . . . 4  |-  ( y  =  B  ->  (
( A G y )  =  ( y G A )  <->  ( A G B )  =  ( B G A ) ) )
106, 9rspc2v 3079 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x )  ->  ( A G B )  =  ( B G A ) ) )
113, 10syl5com 30 . 2  |-  ( G  e.  AbelOp  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) ) )
12113impib 1185 1  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   ran crn 4841  (class class class)co 6091   GrpOpcgr 23673   AbelOpcablo 23768
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
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-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-cnv 4848  df-dm 4850  df-rn 4851  df-iota 5381  df-fv 5426  df-ov 6094  df-ablo 23769
This theorem is referenced by:  ablo32  23773  ablomuldiv  23776  ablodiv32  23779  gxdi  23783  ghablo  23856  rngocom  23879  vccom  23938  nvcom  23999  iscringd  28799
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