Theorem gaid 9454

Description: The trivial action of a group on any set. Each group element corresponds to the identity permutation. (Contributed by Jeff Hankins, 11-Aug-2009.)

Hypothesis

Ref	Expression
gaid.1	|- X = ran G

Assertion

Ref	Expression
gaid	|- ((G e. Grp /\ S e. A) -> <.G, (2nd |` (X X. S))>. e. GrpAct)

Proof of Theorem gaid

Step	Hyp	Ref	Expression
1		simpl 346	. . . 4 |- ((G e. Grp /\ S e. A) -> G e. Grp)
2		f2ndres 5035	. . . . 5 |- (2nd |` (X X. S)):(X X. S)-->S
3	2	a1i 8	. . . 4 |- ((G e. Grp /\ S e. A) -> (2nd |` (X X. S)):(X X. S)-->S)
4		gaid.1	. . . . . . . . . . . . 13 |- X = ran G
5		eqid 1884	. . . . . . . . . . . . 13 |- (Id` G) = (Id` G)
6	4, 5	grpidcl 9343	. . . . . . . . . . . 12 |- (G e. Grp -> (Id` G) e. X)
7	6	3ad2ant1 897	. . . . . . . . . . 11 |- ((G e. Grp /\ S e. A /\ x e. S) -> (Id` G) e. X)
8		simp3 878	. . . . . . . . . . 11 |- ((G e. Grp /\ S e. A /\ x e. S) -> x e. S)
9		opelxpi 4040	. . . . . . . . . . 11 |- (((Id` G) e. X /\ x e. S) -> <.(Id` G), x>. e. (X X. S))
10	7, 8, 9	syl11anc 524	. . . . . . . . . 10 |- ((G e. Grp /\ S e. A /\ x e. S) -> <.(Id` G), x>. e. (X X. S))
11		fvres 4691	. . . . . . . . . 10 |- (<.(Id` G), x>. e. (X X. S) -> ((2nd |` (X X. S))` <.(Id` G), x>.) = (2nd` <.(Id` G), x>.))
12	10, 11	syl 12	. . . . . . . . 9 |- ((G e. Grp /\ S e. A /\ x e. S) -> ((2nd |` (X X. S))` <.(Id` G), x>.) = (2nd` <.(Id` G), x>.))
13		df-opr 4886	. . . . . . . . 9 |- ((Id` G)(2nd |` (X X. S))x) = ((2nd |` (X X. S))` <.(Id` G), x>.)
14	12, 13	syl5eq 1940	. . . . . . . 8 |- ((G e. Grp /\ S e. A /\ x e. S) -> ((Id` G)(2nd |` (X X. S))x) = (2nd` <.(Id` G), x>.))
15		fvex 4689	. . . . . . . . 9 |- (Id` G) e. _V
16		visset 2295	. . . . . . . . 9 |- x e. _V
17	15, 16	op2nd 5027	. . . . . . . 8 |- (2nd` <.(Id` G), x>.) = x
18	14, 17	syl6eq 1944	. . . . . . 7 |- ((G e. Grp /\ S e. A /\ x e. S) -> ((Id` G)(2nd |` (X X. S))x) = x)
19	4	grpcl 9324	. . . . . . . . . . . . . . . . 17 |- ((G e. Grp /\ y e. X /\ z e. X) -> (yGz) e. X)
20	19	3exp 1066	. . . . . . . . . . . . . . . 16 |- (G e. Grp -> (y e. X -> (z e. X -> (yGz) e. X)))
21	20	3ad2ant1 897	. . . . . . . . . . . . . . 15 |- ((G e. Grp /\ S e. A /\ x e. S) -> (y e. X -> (z e. X -> (yGz) e. X)))
22	21	3imp 1061	. . . . . . . . . . . . . 14 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> (yGz) e. X)
23	8	3ad2ant1 897	. . . . . . . . . . . . . 14 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> x e. S)
24		opelxpi 4040	. . . . . . . . . . . . . 14 |- (((yGz) e. X /\ x e. S) -> <.(yGz), x>. e. (X X. S))
25	22, 23, 24	syl11anc 524	. . . . . . . . . . . . 13 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> <.(yGz), x>. e. (X X. S))
26		fvres 4691	. . . . . . . . . . . . 13 |- (<.(yGz), x>. e. (X X. S) -> ((2nd |` (X X. S))` <.(yGz), x>.) = (2nd` <.(yGz), x>.))
27	25, 26	syl 12	. . . . . . . . . . . 12 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> ((2nd |` (X X. S))` <.(yGz), x>.) = (2nd` <.(yGz), x>.))
28		df-opr 4886	. . . . . . . . . . . 12 |- ((yGz)(2nd |` (X X. S))x) = ((2nd |` (X X. S))` <.(yGz), x>.)
29	27, 28	syl5eq 1940	. . . . . . . . . . 11 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> ((yGz)(2nd |` (X X. S))x) = (2nd` <.(yGz), x>.))
30		simp3 878	. . . . . . . . . . . . . . . 16 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> z e. X)
31		opelxpi 4040	. . . . . . . . . . . . . . . 16 |- ((z e. X /\ x e. S) -> <.z, x>. e. (X X. S))
32	30, 23, 31	syl11anc 524	. . . . . . . . . . . . . . 15 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> <.z, x>. e. (X X. S))
33		fvres 4691	. . . . . . . . . . . . . . 15 |- (<.z, x>. e. (X X. S) -> ((2nd |` (X X. S))` <.z, x>.) = (2nd` <.z, x>.))
34	32, 33	syl 12	. . . . . . . . . . . . . 14 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> ((2nd |` (X X. S))` <.z, x>.) = (2nd` <.z, x>.))
35		visset 2295	. . . . . . . . . . . . . . 15 |- z e. _V
36	35, 16	op2nd 5027	. . . . . . . . . . . . . 14 |- (2nd` <.z, x>.) = x
37	34, 36	syl6req 1945	. . . . . . . . . . . . 13 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> x = ((2nd |` (X X. S))` <.z, x>.))
38		df-opr 4886	. . . . . . . . . . . . 13 |- (z(2nd |` (X X. S))x) = ((2nd |` (X X. S))` <.z, x>.)
39	37, 38	syl6eqr 1946	. . . . . . . . . . . 12 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> x = (z(2nd |` (X X. S))x))
40		oprex 4907	. . . . . . . . . . . . 13 |- (yGz) e. _V
41	40, 16	op2nd 5027	. . . . . . . . . . . 12 |- (2nd` <.(yGz), x>.) = x
42		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
43		oprex 4907	. . . . . . . . . . . . 13 |- (z(2nd |` (X X. S))x) e. _V
44	42, 43	op2nd 5027	. . . . . . . . . . . 12 |- (2nd` <.y, (z(2nd |` (X X. S))x)>.) = (z(2nd |` (X X. S))x)
45	39, 41, 44	3eqtr4g 1953	. . . . . . . . . . 11 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> (2nd` <.(yGz), x>.) = (2nd` <.y, (z(2nd |` (X X. S))x)>.))
46		simp2 877	. . . . . . . . . . . . . 14 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> y e. X)
47	34, 38	syl5eq 1940	. . . . . . . . . . . . . . 15 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> (z(2nd |` (X X. S))x) = (2nd` <.z, x>.))
48	23, 36	syl5eqel 1975	. . . . . . . . . . . . . . 15 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> (2nd` <.z, x>.) e. S)
49	47, 48	eqeltrd 1971	. . . . . . . . . . . . . 14 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> (z(2nd |` (X X. S))x) e. S)
50		opelxpi 4040	. . . . . . . . . . . . . 14 |- ((y e. X /\ (z(2nd |` (X X. S))x) e. S) -> <.y, (z(2nd |` (X X. S))x)>. e. (X X. S))
51	46, 49, 50	syl11anc 524	. . . . . . . . . . . . 13 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> <.y, (z(2nd |` (X X. S))x)>. e. (X X. S))
52		fvres 4691	. . . . . . . . . . . . 13 |- (<.y, (z(2nd |` (X X. S))x)>. e. (X X. S) -> ((2nd |` (X X. S))` <.y, (z(2nd |` (X X. S))x)>.) = (2nd` <.y, (z(2nd |` (X X. S))x)>.))
53	51, 52	syl 12	. . . . . . . . . . . 12 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> ((2nd |` (X X. S))` <.y, (z(2nd |` (X X. S))x)>.) = (2nd` <.y, (z(2nd |` (X X. S))x)>.))
54		df-opr 4886	. . . . . . . . . . . 12 |- (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)) = ((2nd |` (X X. S))` <.y, (z(2nd |` (X X. S))x)>.)
55	53, 54	syl5req 1941	. . . . . . . . . . 11 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> (2nd` <.y, (z(2nd |` (X X. S))x)>.) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))
56	29, 45, 55	3eqtrd 1929	. . . . . . . . . 10 |- (((G e. Grp /\ S e. A /\ x e. S) /\ y e. X /\ z e. X) -> ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))
57	56	3exp 1066	. . . . . . . . 9 |- ((G e. Grp /\ S e. A /\ x e. S) -> (y e. X -> (z e. X -> ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))))
58	57	r19.21adv 2181	. . . . . . . 8 |- ((G e. Grp /\ S e. A /\ x e. S) -> (y e. X -> A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x))))
59	58	r19.21aiv 2175	. . . . . . 7 |- ((G e. Grp /\ S e. A /\ x e. S) -> A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))
60	18, 59	jca 310	. . . . . 6 |- ((G e. Grp /\ S e. A /\ x e. S) -> (((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x))))
61	60	3expia 1069	. . . . 5 |- ((G e. Grp /\ S e. A) -> (x e. S -> (((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))))
62	61	r19.21aiv 2175	. . . 4 |- ((G e. Grp /\ S e. A) -> A.x e. S (((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x))))
63	1, 3, 62	3jca 1050	. . 3 |- ((G e. Grp /\ S e. A) -> (G e. Grp /\ (2nd |` (X X. S)):(X X. S)-->S /\ A.x e. S (((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))))
64	4	grpn0 9326	. . . . . 6 |- (G e. Grp -> X =/= (/))
65		fo2ndres 5037	. . . . . 6 |- (X =/= (/) -> (2nd |` (X X. S)):(X X. S)-onto->S)
66		forn 4620	. . . . . 6 |- ((2nd |` (X X. S)):(X X. S)-onto->S -> ran (2nd |` (X X. S)) = S)
67	64, 65, 66	3syl 24	. . . . 5 |- (G e. Grp -> ran (2nd |` (X X. S)) = S)
68	67	adantr 425	. . . 4 |- ((G e. Grp /\ S e. A) -> ran (2nd |` (X X. S)) = S)
69		xpeq2 4017	. . . . . 6 |- (ran (2nd |` (X X. S)) = S -> (X X. ran (2nd |` (X X. S))) = (X X. S))
70		feq23 4554	. . . . . 6 |- (((X X. ran (2nd |` (X X. S))) = (X X. S) /\ ran (2nd |` (X X. S)) = S) -> ((2nd |` (X X. S)):(X X. ran (2nd |` (X X. S)))-->ran (2nd |` (X X. S)) <-> (2nd |` (X X. S)):(X X. S)-->S))
71	69, 70	mpancom 769	. . . . 5 |- (ran (2nd |` (X X. S)) = S -> ((2nd |` (X X. S)):(X X. ran (2nd |` (X X. S)))-->ran (2nd |` (X X. S)) <-> (2nd |` (X X. S)):(X X. S)-->S))
72		raleq 2266	. . . . 5 |- (ran (2nd |` (X X. S)) = S -> (A.x e. ran (2nd |` (X X. S))(((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x))) <-> A.x e. S (((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))))
73	71, 72	3anbi23d 1171	. . . 4 |- (ran (2nd |` (X X. S)) = S -> ((G e. Grp /\ (2nd |` (X X. S)):(X X. ran (2nd |` (X X. S)))-->ran (2nd |` (X X. S)) /\ A.x e. ran (2nd |` (X X. S))(((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))) <-> (G e. Grp /\ (2nd |` (X X. S)):(X X. S)-->S /\ A.x e. S (((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x))))))
74	68, 73	syl 12	. . 3 |- ((G e. Grp /\ S e. A) -> ((G e. Grp /\ (2nd |` (X X. S)):(X X. ran (2nd |` (X X. S)))-->ran (2nd |` (X X. S)) /\ A.x e. ran (2nd |` (X X. S))(((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))) <-> (G e. Grp /\ (2nd |` (X X. S)):(X X. S)-->S /\ A.x e. S (((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x))))))
75	63, 74	mpbird 213	. 2 |- ((G e. Grp /\ S e. A) -> (G e. Grp /\ (2nd |` (X X. S)):(X X. ran (2nd |` (X X. S)))-->ran (2nd |` (X X. S)) /\ A.x e. ran (2nd |` (X X. S))(((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x)))))
76		fex 4595	. . . 4 |- (((2nd |` (X X. S)):(X X. S)-->S /\ (X X. S) e. _V) -> (2nd |` (X X. S)) e. _V)
77		xpexg 4095	. . . . 5 |- ((X e. _V /\ S e. A) -> (X X. S) e. _V)
78		rnexg 4207	. . . . . 6 |- (G e. Grp -> ran G e. _V)
79	78, 4	syl5eqel 1975	. . . . 5 |- (G e. Grp -> X e. _V)
80	77, 79	sylan 497	. . . 4 |- ((G e. Grp /\ S e. A) -> (X X. S) e. _V)
81	76, 2, 80	sylancr 526	. . 3 |- ((G e. Grp /\ S e. A) -> (2nd |` (X X. S)) e. _V)
82		eqid 1884	. . . 4 |- ran (2nd |` (X X. S)) = ran (2nd |` (X X. S))
83	4, 82, 5	isga2 9452	. . 3 |- ((2nd |` (X X. S)) e. _V -> (<.G, (2nd |` (X X. S))>. e. GrpAct <-> (G e. Grp /\ (2nd |` (X X. S)):(X X. ran (2nd |` (X X. S)))-->ran (2nd |` (X X. S)) /\ A.x e. ran (2nd |` (X X. S))(((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x))))))
84	81, 83	syl 12	. 2 |- ((G e. Grp /\ S e. A) -> (<.G, (2nd |` (X X. S))>. e. GrpAct <-> (G e. Grp /\ (2nd |` (X X. S)):(X X. ran (2nd |` (X X. S)))-->ran (2nd |` (X X. S)) /\ A.x e. ran (2nd |` (X X. S))(((Id` G)(2nd |` (X X. S))x) = x /\ A.y e. X A.z e. X ((yGz)(2nd |` (X X. S))x) = (y(2nd |` (X X. S))(z(2nd |` (X X. S))x))))))
85	75, 84	mpbird 213	1 |- ((G e. Grp /\ S e. A) -> <.G, (2nd |` (X X. S))>. e. GrpAct)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292  (/)c0 2875  <.cop 3046   X. cxp 3984  ran crn 3987   |` cres 3988  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  2ndc2nd 5019  Grpcgr 9311  Idcgi 9312  GrpActcga 9447

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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  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-2nd 5021  df-grp 9316  df-gid 9317  df-ga 9448

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