Theorem ablsn 8110

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 8109	. 2 |- {<.<.A, A>., A>.} e. Grp
3		opex 2779	. . . . 5 |- <.A, A>. e. V
4	3, 1	f1osn 3716	. . . 4 |- {<.<.A, A>., A>.}:{<.A, A>.}-1-1-onto->{A}
5		f1of 3686	. . . 4 |- ({<.<.A, A>., A>.}:{<.A, A>.}-1-1-onto->{A} -> {<.<.A, A>., A>.}:{<.A, A>.}-->{A})
6	4, 5	ax-mp 7	. . 3 |- {<.<.A, A>., A>.}:{<.A, A>.}-->{A}
7	1, 1	xpsn 3832	. . . 4 |- ({A} X. {A}) = {<.A, A>.}
8		feq2 3618	. . . 4 |- (({A} X. {A}) = {<.A, A>.} -> ({<.<.A, A>., A>.}:({A} X. {A})-->{A} <-> {<.<.A, A>., A>.}:{<.A, A>.}-->{A}))
9	7, 8	ax-mp 7	. . 3 |- ({<.<.A, A>., A>.}:({A} X. {A})-->{A} <-> {<.<.A, A>., A>.}:{<.A, A>.}-->{A})
10	6, 9	mpbir 190	. 2 |- {<.<.A, A>., A>.}:({A} X. {A})-->{A}
11		opreq12 3967	. . . 4 |- ((x = A /\ y = A) -> (x{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}A))
12		opreq2 3966	. . . . 5 |- (x = A -> (y{<.<.A, A>., A>.}x) = (y{<.<.A, A>., A>.}A))
13		opreq1 3965	. . . . 5 |- (y = A -> (y{<.<.A, A>., A>.}A) = (A{<.<.A, A>., A>.}A))
14	12, 13	sylan9eq 1526	. . . 4 |- ((x = A /\ y = A) -> (y{<.<.A, A>., A>.}x) = (A{<.<.A, A>., A>.}A))
15	11, 14	eqtr4d 1509	. . 3 |- ((x = A /\ y = A) -> (x{<.<.A, A>., A>.}y) = (y{<.<.A, A>., A>.}x))
16		elsn 2419	. . 3 |- (x e. {A} <-> x = A)
17		elsn 2419	. . 3 |- (y e. {A} <-> y = A)
18	15, 16, 17	syl2anb 455	. 2 |- ((x e. {A} /\ y e. {A}) -> (x{<.<.A, A>., A>.}y) = (y{<.<.A, A>., A>.}x))
19	2, 10, 18	isabliOLD 8090	1 |- {<.<.A, A>., A>.} e. Abel

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1809  {csn 2407  <.cop 2409   X. cxp 3165  -->wf 3175  -1-1-onto->wf1o 3178  (class class class)co 3960  Abelcabl 8083

This theorem is referenced by:  ringsn 8148

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195  df-opr 3962  df-grp 8020  df-abl 8084

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