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Theorem ablocom 25701
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 25699 . . . 4  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
32simprbi 462 . . 3  |-  ( G  e.  AbelOp  ->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) )
4 oveq1 6285 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
5 oveq2 6286 . . . . 5  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
64, 5eqeq12d 2424 . . . 4  |-  ( x  =  A  ->  (
( x G y )  =  ( y G x )  <->  ( A G y )  =  ( y G A ) ) )
7 oveq2 6286 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
8 oveq1 6285 . . . . 5  |-  ( y  =  B  ->  (
y G A )  =  ( B G A ) )
97, 8eqeq12d 2424 . . . 4  |-  ( y  =  B  ->  (
( A G y )  =  ( y G A )  <->  ( A G B )  =  ( B G A ) ) )
106, 9rspc2v 3169 . . 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 28 . 2  |-  ( G  e.  AbelOp  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) ) )
12113impib 1195 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   ran crn 4824  (class class class)co 6278   GrpOpcgr 25602   AbelOpcablo 25697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-cnv 4831  df-dm 4833  df-rn 4834  df-iota 5533  df-fv 5577  df-ov 6281  df-ablo 25698
This theorem is referenced by:  ablo32  25702  ablomuldiv  25705  ablodiv32  25708  gxdi  25712  ghabloOLD  25785  rngocom  25808  vccom  25867  nvcom  25928  iscringd  31678
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