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Theorem ablocom 24960
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 24958 . . . 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 6289 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
5 oveq2 6290 . . . . 5  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
64, 5eqeq12d 2489 . . . 4  |-  ( x  =  A  ->  (
( x G y )  =  ( y G x )  <->  ( A G y )  =  ( y G A ) ) )
7 oveq2 6290 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
8 oveq1 6289 . . . . 5  |-  ( y  =  B  ->  (
y G A )  =  ( B G A ) )
97, 8eqeq12d 2489 . . . 4  |-  ( y  =  B  ->  (
( A G y )  =  ( y G A )  <->  ( A G B )  =  ( B G A ) ) )
106, 9rspc2v 3223 . . 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 1194 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   ran crn 5000  (class class class)co 6282   GrpOpcgr 24861   AbelOpcablo 24956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010  df-iota 5549  df-fv 5594  df-ov 6285  df-ablo 24957
This theorem is referenced by:  ablo32  24961  ablomuldiv  24964  ablodiv32  24967  gxdi  24971  ghablo  25044  rngocom  25067  vccom  25126  nvcom  25187  iscringd  29997
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