Theorem ablsn 8121

Description: The Abelian group operation for the singleton group.

Hypothesis

Ref	Expression
ablsn.1	|- A e. V

Assertion

Ref	Expression
ablsn	|- {<.<.A, A>., A>.} e. Abel

Proof of Theorem ablsn

Step	Hyp	Ref	Expression
1		ablsn.1	. . 3 |- A e. V
2	1	grpsn 8120	. 2 |- {<.<.A, A>., A>.} e. Grp
3		dmsnop 3334	. . 3 |- dom {<.<.A, A>., A>.} = {<.A, A>.}
4	1, 1	xpsn 3841	. . 3 |- ({A} X. {A}) = {<.A, A>.}
5	3, 4	eqtr4 1501	. 2 |- dom {<.<.A, A>., A>.} = ({A} X. {A})
6		opreq12 3976	. . . 4 |- ((x = A /\ y = A) -> (x{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}A))
7		opreq2 3975	. . . . 5 |- (x = A -> (y{<.<.A, A>., A>.}x) = (y{<.<.A, A>., A>.}A))
8		opreq1 3974	. . . . 5 |- (y = A -> (y{<.<.A, A>., A>.}A) = (A{<.<.A, A>., A>.}A))
9	7, 8	sylan9eq 1530	. . . 4 |- ((x = A /\ y = A) -> (y{<.<.A, A>., A>.}x) = (A{<.<.A, A>., A>.}A))
10	6, 9	eqtr4d 1513	. . 3 |- ((x = A /\ y = A) -> (x{<.<.A, A>., A>.}y) = (y{<.<.A, A>., A>.}x))
11		elsn 2425	. . 3 |- (x e. {A} <-> x = A)
12		elsn 2425	. . 3 |- (y e. {A} <-> y = A)
13	10, 11, 12	syl2anb 457	. 2 |- ((x e. {A} /\ y e. {A}) -> (x{<.<.A, A>., A>.}y) = (y{<.<.A, A>., A>.}x))
14	2, 5, 13	isabli 8102	1 |- {<.<.A, A>., A>.} e. Abel

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  {csn 2413  <.cop 2415   X. cxp 3174  dom cdm 3176  (class class class)co 3969  Abelcabl 8095

This theorem is referenced by:  ringsn 8159

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-opr 3971  df-grp 8034  df-abl 8096

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