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Theorem ablocom 21873
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 21871 . . . 4  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
32simprbi 451 . . 3  |-  ( G  e.  AbelOp  ->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) )
4 oveq1 6088 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
5 oveq2 6089 . . . . 5  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
64, 5eqeq12d 2450 . . . 4  |-  ( x  =  A  ->  (
( x G y )  =  ( y G x )  <->  ( A G y )  =  ( y G A ) ) )
7 oveq2 6089 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
8 oveq1 6088 . . . . 5  |-  ( y  =  B  ->  (
y G A )  =  ( B G A ) )
97, 8eqeq12d 2450 . . . 4  |-  ( y  =  B  ->  (
( A G y )  =  ( y G A )  <->  ( A G B )  =  ( B G A ) ) )
106, 9rspc2v 3058 . . 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 1151 1  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   ran crn 4879  (class class class)co 6081   GrpOpcgr 21774   AbelOpcablo 21869
This theorem is referenced by:  ablo32  21874  ablomuldiv  21877  ablodiv32  21880  gxdi  21884  ghablo  21957  rngocom  21980  vccom  22039  nvcom  22100  iscringd  26609
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889  df-iota 5418  df-fv 5462  df-ov 6084  df-ablo 21870
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