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Theorem ablosn 25011
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 24879 . 2  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
31dmsnop 5473 . . 3  |-  dom  { <. <. A ,  A >. ,  A >. }  =  { <. A ,  A >. }
41, 1xpsn 6054 . . 3  |-  ( { A }  X.  { A } )  =  { <. A ,  A >. }
53, 4eqtr4i 2492 . 2  |-  dom  { <. <. A ,  A >. ,  A >. }  =  ( { A }  X.  { A } )
6 elsn 4034 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
7 elsn 4034 . . 3  |-  ( y  e.  { A }  <->  y  =  A )
8 oveq12 6284 . . . 4  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
9 oveq2 6283 . . . . 5  |-  ( x  =  A  ->  (
y { <. <. A ,  A >. ,  A >. } x )  =  ( y { <. <. A ,  A >. ,  A >. } A ) )
10 oveq1 6282 . . . . 5  |-  ( y  =  A  ->  (
y { <. <. A ,  A >. ,  A >. } A )  =  ( A { <. <. A ,  A >. ,  A >. } A ) )
119, 10sylan9eq 2521 . . . 4  |-  ( ( x  =  A  /\  y  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } x )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
128, 11eqtr4d 2504 . . 3  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( y {
<. <. A ,  A >. ,  A >. } x
) )
136, 7, 12syl2anb 479 . 2  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x { <. <. A ,  A >. ,  A >. } y )  =  ( y { <. <. A ,  A >. ,  A >. } x ) )
142, 5, 13isabloi 24952 1  |-  { <. <. A ,  A >. ,  A >. }  e.  AbelOp
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106   {csn 4020   <.cop 4026    X. cxp 4990   dom cdm 4992  (class class class)co 6275   AbelOpcablo 24945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-grpo 24855  df-ablo 24946
This theorem is referenced by:  rngosn  25068
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