Theorem grpsn 8243

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 2802	. 2 |- {A} e. V
2		opex 2835	. . . . 5 |- <.A, A>. e. V
3		grpsn.1	. . . . 5 |- A e. V
4	2, 3	f1osn 3795	. . . 4 |- {<.<.A, A>., A>.}:{<.A, A>.}-1-1-onto->{A}
5		f1of 3765	. . . 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 3911	. . . 4 |- ({A} X. {A}) = {<.A, A>.}
8		feq2 3696	. . . 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 188	. 2 |- {<.<.A, A>., A>.}:({A} X. {A})-->{A}
11		opreq2 4045	. . . . . 6 |- (z = A -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}A))
12		opreq1 4044	. . . . . . . . 9 |- (x = A -> (x{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}y))
13		opreq2 4045	. . . . . . . . . 10 |- (y = A -> (A{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}A))
14		df-opr 4041	. . . . . . . . . . 11 |- (A{<.<.A, A>., A>.}A) = ({<.<.A, A>., A>.}` <.A, A>.)
15	2, 3	fvsn 3870	. . . . . . . . . . 11 |- ({<.<.A, A>., A>.}` <.A, A>.) = A
16	14, 15	eqtri 1532	. . . . . . . . . 10 |- (A{<.<.A, A>., A>.}A) = A
17	13, 16	syl6eq 1560	. . . . . . . . 9 |- (y = A -> (A{<.<.A, A>., A>.}y) = A)
18	12, 17	sylan9eq 1564	. . . . . . . 8 |- ((x = A /\ y = A) -> (x{<.<.A, A>., A>.}y) = A)
19	18	opreq1d 4051	. . . . . . 7 |- ((x = A /\ y = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}A) = (A{<.<.A, A>., A>.}A))
20	19, 16	syl6eq 1560	. . . . . 6 |- ((x = A /\ y = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}A) = A)
21	11, 20	sylan9eqr 1566	. . . . 5 |- (((x = A /\ y = A) /\ z = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = A)
22	21	3impa 831	. . . 4 |- ((x = A /\ y = A /\ z = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = A)
23		opreq1 4044	. . . . . 6 |- (x = A -> (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = (A{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)))
24		opreq1 4044	. . . . . . . . 9 |- (y = A -> (y{<.<.A, A>., A>.}z) = (A{<.<.A, A>., A>.}z))
25		opreq2 4045	. . . . . . . . . 10 |- (z = A -> (A{<.<.A, A>., A>.}z) = (A{<.<.A, A>., A>.}A))
26	25, 16	syl6eq 1560	. . . . . . . . 9 |- (z = A -> (A{<.<.A, A>., A>.}z) = A)
27	24, 26	sylan9eq 1564	. . . . . . . 8 |- ((y = A /\ z = A) -> (y{<.<.A, A>., A>.}z) = A)
28	27	opreq2d 4052	. . . . . . 7 |- ((y = A /\ z = A) -> (A{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = (A{<.<.A, A>., A>.}A))
29	28, 16	syl6eq 1560	. . . . . 6 |- ((y = A /\ z = A) -> (A{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = A)
30	23, 29	sylan9eq 1564	. . . . 5 |- ((x = A /\ (y = A /\ z = A)) -> (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = A)
31	30	3impb 832	. . . 4 |- ((x = A /\ y = A /\ z = A) -> (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = A)
32	22, 31	eqtr4d 1547	. . 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 2466	. . 3 |- (x e. {A} <-> x = A)
34		elsn 2466	. . 3 |- (y e. {A} <-> y = A)
35		elsn 2466	. . 3 |- (z e. {A} <-> z = A)
36	32, 33, 34, 35	syl3anb 872	. 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 2480	. 2 |- A e. {A}
38		opreq2 4045	. . . 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 1569	. . 3 |- (x = A -> (A{<.<.A, A>., A>.}x) = x)
41	33, 40	sylbi 197	. 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 1560	. . 3 |- (x = A -> (A{<.<.A, A>., A>.}x) = A)
44	33, 43	sylbi 197	. 2 |- (x e. {A} -> (A{<.<.A, A>., A>.}x) = A)
45	1, 10, 36, 37, 41, 42, 44	isgrpi 8162	1 |- {<.<.A, A>., A>.} e. Grp

Syntax hints:   <-> wb 144   /\ wa 221   /\ w3a 778   = wceq 988   e. wcel 990  Vcvv 1849  {csn 2454  <.cop 2456   X. cxp 3223  -->wf 3233  -1-1-onto->wf1o 3236  ` cfv 3237  (class class class)co 4039  Grpcgr 8153

This theorem is referenced by:  ablsn 8244  ghomsn 10509  ghomgrplem 10510  cayleythlem 10534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-rep 2744  ax-sep 2754  ax-nul 2761  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-reu 1689  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-fv 3253  df-opr 4041  df-grp 8157

Copyright terms: Public domain