Theorem grpsn 9340

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 3492	. 2 |- {A} e. _V
2		opex 3527	. . . . 5 |- <.A, A>. e. _V
3		grpsn.1	. . . . 5 |- A e. _V
4	2, 3	f1osn 4674	. . . 4 |- {<.<.A, A>., A>.}:{<.A, A>.}-1-1-onto->{A}
5		f1of 4635	. . . 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 4808	. . . 4 |- ({A} X. {A}) = {<.A, A>.}
8	7	feq2i 4559	. . 3 |- ({<.<.A, A>., A>.}:({A} X. {A})-->{A} <-> {<.<.A, A>., A>.}:{<.A, A>.}-->{A})
9	6, 8	mpbir 207	. 2 |- {<.<.A, A>., A>.}:({A} X. {A})-->{A}
10		opreq2 4890	. . . . . 6 |- (z = A -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}A))
11		opreq1 4889	. . . . . . . . 9 |- (x = A -> (x{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}y))
12		opreq2 4890	. . . . . . . . . 10 |- (y = A -> (A{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}A))
13		df-opr 4886	. . . . . . . . . . 11 |- (A{<.<.A, A>., A>.}A) = ({<.<.A, A>., A>.}` <.A, A>.)
14	2, 3	fvsn 4758	. . . . . . . . . . 11 |- ({<.<.A, A>., A>.}` <.A, A>.) = A
15	13, 14	eqtri 1908	. . . . . . . . . 10 |- (A{<.<.A, A>., A>.}A) = A
16	12, 15	syl6eq 1944	. . . . . . . . 9 |- (y = A -> (A{<.<.A, A>., A>.}y) = A)
17	11, 16	sylan9eq 1948	. . . . . . . 8 |- ((x = A /\ y = A) -> (x{<.<.A, A>., A>.}y) = A)
18	17	opreq1d 4897	. . . . . . 7 |- ((x = A /\ y = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}A) = (A{<.<.A, A>., A>.}A))
19	18, 15	syl6eq 1944	. . . . . 6 |- ((x = A /\ y = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}A) = A)
20	10, 19	sylan9eqr 1951	. . . . 5 |- (((x = A /\ y = A) /\ z = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = A)
21	20	3impa 1062	. . . 4 |- ((x = A /\ y = A /\ z = A) -> ((x{<.<.A, A>., A>.}y){<.<.A, A>., A>.}z) = A)
22		opreq1 4889	. . . . . 6 |- (x = A -> (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = (A{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)))
23		opreq1 4889	. . . . . . . . 9 |- (y = A -> (y{<.<.A, A>., A>.}z) = (A{<.<.A, A>., A>.}z))
24		opreq2 4890	. . . . . . . . . 10 |- (z = A -> (A{<.<.A, A>., A>.}z) = (A{<.<.A, A>., A>.}A))
25	24, 15	syl6eq 1944	. . . . . . . . 9 |- (z = A -> (A{<.<.A, A>., A>.}z) = A)
26	23, 25	sylan9eq 1948	. . . . . . . 8 |- ((y = A /\ z = A) -> (y{<.<.A, A>., A>.}z) = A)
27	26	opreq2d 4898	. . . . . . 7 |- ((y = A /\ z = A) -> (A{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = (A{<.<.A, A>., A>.}A))
28	27, 15	syl6eq 1944	. . . . . 6 |- ((y = A /\ z = A) -> (A{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = A)
29	22, 28	sylan9eq 1948	. . . . 5 |- ((x = A /\ (y = A /\ z = A)) -> (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = A)
30	29	3impb 1063	. . . 4 |- ((x = A /\ y = A /\ z = A) -> (x{<.<.A, A>., A>.} (y{<.<.A, A>., A>.}z)) = A)
31	21, 30	eqtr4d 1928	. . 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)))
32		elsn 3058	. . 3 |- (x e. {A} <-> x = A)
33		elsn 3058	. . 3 |- (y e. {A} <-> y = A)
34		elsn 3058	. . 3 |- (z e. {A} <-> z = A)
35	31, 32, 33, 34	syl3anb 1140	. 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)))
36	3	snid 3069	. 2 |- A e. {A}
37		opreq2 4890	. . . 4 |- (x = A -> (A{<.<.A, A>., A>.}x) = (A{<.<.A, A>., A>.}A))
38		id 73	. . . 4 |- (x = A -> x = A)
39	15, 37, 38	3eqtr4a 1954	. . 3 |- (x = A -> (A{<.<.A, A>., A>.}x) = x)
40	32, 39	sylbi 216	. 2 |- (x e. {A} -> (A{<.<.A, A>., A>.}x) = x)
41	36	a1i 8	. 2 |- (x e. {A} -> A e. {A})
42	37, 15	syl6eq 1944	. . 3 |- (x = A -> (A{<.<.A, A>., A>.}x) = A)
43	32, 42	sylbi 216	. 2 |- (x e. {A} -> (A{<.<.A, A>., A>.}x) = A)
44	1, 9, 35, 36, 40, 41, 43	isgrpi 9322	1 |- {<.<.A, A>., A>.} e. Grp

Syntax hints:   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044  <.cop 3046   X. cxp 3984  -->wf 3994  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Grpcgr 9311

This theorem is referenced by:  grpidval 9342  ablsn 9433  zrdivrng 10418  ghomsn 13631  ghomgrplem 13632  cayleythlem 13645  idtrgrp 14978  invtrgrp 14979  extopgrp 14980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-grp 9316

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