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Theorem ablosn 21935
Description: The Abelian group operation for the singleton group. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
ablsn.1  |-  A  e. 
_V
Assertion
Ref Expression
ablosn  |-  { <. <. A ,  A >. ,  A >. }  e.  AbelOp

Proof of Theorem ablosn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablsn.1 . . 3  |-  A  e. 
_V
21grposn 21803 . 2  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
31dmsnop 5344 . . 3  |-  dom  { <. <. A ,  A >. ,  A >. }  =  { <. A ,  A >. }
41, 1xpsn 5910 . . 3  |-  ( { A }  X.  { A } )  =  { <. A ,  A >. }
53, 4eqtr4i 2459 . 2  |-  dom  { <. <. A ,  A >. ,  A >. }  =  ( { A }  X.  { A } )
6 elsn 3829 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
7 elsn 3829 . . 3  |-  ( y  e.  { A }  <->  y  =  A )
8 oveq12 6090 . . . 4  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
9 oveq2 6089 . . . . 5  |-  ( x  =  A  ->  (
y { <. <. A ,  A >. ,  A >. } x )  =  ( y { <. <. A ,  A >. ,  A >. } A ) )
10 oveq1 6088 . . . . 5  |-  ( y  =  A  ->  (
y { <. <. A ,  A >. ,  A >. } A )  =  ( A { <. <. A ,  A >. ,  A >. } A ) )
119, 10sylan9eq 2488 . . . 4  |-  ( ( x  =  A  /\  y  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } x )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
128, 11eqtr4d 2471 . . 3  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( y {
<. <. A ,  A >. ,  A >. } x
) )
136, 7, 12syl2anb 466 . 2  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x { <. <. A ,  A >. ,  A >. } y )  =  ( y { <. <. A ,  A >. ,  A >. } x ) )
142, 5, 13isabloi 21876 1  |-  { <. <. A ,  A >. ,  A >. }  e.  AbelOp
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814   <.cop 3817    X. cxp 4876   dom cdm 4878  (class class class)co 6081   AbelOpcablo 21869
This theorem is referenced by:  rngosn  21992
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-grpo 21779  df-ablo 21870
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