Theorem grpsn 8120

Description: The group operation for the singleton group.

Hypothesis

Ref	Expression
grpsn.1	|- A e. V

Assertion

Ref	Expression
grpsn	|- {<.<.A, A>., A>.} e. Grp

Proof of Theorem grpsn

Step	Hyp	Ref	Expression
1		snex 2756	. 2 |- {A} e. V
2		opex 2788	. . . . 5 |- <.A, A>. e. V
3		grpsn.1	. . . . 5 |- A e. V
4	2, 3	f1osn 3725	. . . 4 |- {<.<.A, A>., A>.}:{<.A, A>.}-1-1-onto->{A}
5		f1of 3695	. . . 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	3, 3	xpsn 3841	. . . 4 |- ({A} X. {A}) = {<.A, A>.}
8		feq2 3627	. . . 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		opreq2 3975	. . . . . 6 |- (z = A -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}A))
12		opreq1 3974	. . . . . . . . 9 |- (x = A -> (x{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}y))
13		opreq2 3975	. . . . . . . . . 10 |- (y = A -> (A{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}A))
14		df-opr 3971	. . . . . . . . . . 11 |- (A{<.<.A, A>., A>.}A) = ({<.<.A, A>., A>.}` <.A, A>.)
15	2, 3	fvsn 3800	. . . . . . . . . . 11 |- ({<.<.A, A>., A>.}` <.A, A>.) = A
16	14, 15	eqtr 1498	. . . . . . . . . 10 |- (A{<.<.A, A>., A>.}A) = A
17	13, 16	syl6eq 1526	. . . . . . . . 9 |- (y = A -> (A{<.<.A, A>., A>.}y) = A)
18	12, 17	sylan9eq 1530	. . . . . . . 8 |- ((x = A /\ y = A) -> (x{<.<.A, A>., A>.}y) = A)
19	18	opreq1d 3981	. . . . . . 7 |- ((x = A /\ y = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}A) = (A{<.<.A, A>., A>.}A))
20	19, 16	syl6eq 1526	. . . . . 6 |- ((x = A /\ y = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}A) = A)
21	11, 20	sylan9eqr 1532	. . . . 5 |- (((x = A /\ y = A) /\ z = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = A)
22	21	3impa 830	. . . 4 |- ((x = A /\ y = A /\ z = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = A)
23		opreq1 3974	. . . . . 6 |- (x = A -> (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = (A{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)))
24		opreq1 3974	. . . . . . . . 9 |- (y = A -> (y{<.<.A, A>., A>.}z) = (A{<.<.A, A>., A>.}z))
25		opreq2 3975	. . . . . . . . . 10 |- (z = A -> (A{<.<.A, A>., A>.}z) = (A{<.<.A, A>., A>.}A))
26	25, 16	syl6eq 1526	. . . . . . . . 9 |- (z = A -> (A{<.<.A, A>., A>.}z) = A)
27	24, 26	sylan9eq 1530	. . . . . . . 8 |- ((y = A /\ z = A) -> (y{<.<.A, A>., A>.}z) = A)
28	27	opreq2d 3982	. . . . . . 7 |- ((y = A /\ z = A) -> (A{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = (A{<.<.A, A>., A>.}A))
29	28, 16	syl6eq 1526	. . . . . 6 |- ((y = A /\ z = A) -> (A{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = A)
30	23, 29	sylan9eq 1530	. . . . 5 |- ((x = A /\ (y = A /\ z = A)) -> (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = A)
31	30	3impb 831	. . . 4 |- ((x = A /\ y = A /\ z = A) -> (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = A)
32	22, 31	eqtr4d 1513	. . 3 |- ((x = A /\ y = A /\ z = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)))
33		elsn 2425	. . 3 |- (x e. {A} <-> x = A)
34		elsn 2425	. . 3 |- (y e. {A} <-> y = A)
35		elsn 2425	. . 3 |- (z e. {A} <-> z = A)
36	32, 33, 34, 35	syl3anb 871	. 2 |- ((x e. {A} /\ y e. {A} /\ z e. {A}) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)))
37	3	snid 2439	. 2 |- A e. {A}
38		opreq2 3975	. . . 4 |- (x = A -> (A{<.<.A, A>., A>.}x) = (A{<.<.A, A>., A>.}A))
39		id 59	. . . 4 |- (x = A -> x = A)
40	16, 38, 39	3eqtr4a 1535	. . 3 |- (x = A -> (A{<.<.A, A>., A>.}x) = x)
41	33, 40	sylbi 199	. 2 |- (x e. {A} -> (A{<.<.A, A>., A>.}x) = x)
42	37	a1i 8	. 2 |- (x e. {A} -> A e. {A})
43	38, 16	syl6eq 1526	. . 3 |- (x = A -> (A{<.<.A, A>., A>.}x) = A)
44	33, 43	sylbi 199	. 2 |- (x e. {A} -> (A{<.<.A, A>., A>.}x) = A)
45	1, 10, 36, 37, 41, 42, 44	isgrpi 8039	1 |- {<.<.A, A>., A>.} e. Grp

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  Vcvv 1814  {csn 2413  <.cop 2415   X. cxp 3174  -->wf 3184  -1-1-onto->wf1o 3187  ` cfv 3188  (class class class)co 3969  Grpcgr 8030

This theorem is referenced by:  ablsn 8121  ghomsn 10383  ghomgrplem 10384  cayleythlem 10408

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-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

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