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Theorem ablocom 20968
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 20966 . . . 4  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
32simprbi 450 . . 3  |-  ( G  e.  AbelOp  ->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) )
4 oveq1 5881 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
5 oveq2 5882 . . . . 5  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
64, 5eqeq12d 2310 . . . 4  |-  ( x  =  A  ->  (
( x G y )  =  ( y G x )  <->  ( A G y )  =  ( y G A ) ) )
7 oveq2 5882 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
8 oveq1 5881 . . . . 5  |-  ( y  =  B  ->  (
y G A )  =  ( B G A ) )
97, 8eqeq12d 2310 . . . 4  |-  ( y  =  B  ->  (
( A G y )  =  ( y G A )  <->  ( A G B )  =  ( B G A ) ) )
106, 9rspc2v 2903 . . 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 26 . 2  |-  ( G  e.  AbelOp  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) ) )
12113impib 1149 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   ran crn 4706  (class class class)co 5874   GrpOpcgr 20869   AbelOpcablo 20964
This theorem is referenced by:  ablo32  20969  ablomuldiv  20972  ablodiv32  20975  gxdi  20979  ghablo  21052  rngocom  21075  vccom  21132  nvcom  21193  abloinvop  25456  fprodneg  25481  addvecom  25569  iscringd  26727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279  df-ov 5877  df-ablo 20965
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