Theorem issubgi 8122

Description: Properties that determine a subgroup. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypotheses

Ref	Expression
issubgi.1	|- G e. Grp
issubgi.2	|- X = ran G
issubgi.3	|- U = (Id` G)
issubgi.4	|- N = (inv` G)
issubgi.5	|- Y (_ X
issubgi.6	|- H = (G |` (Y X. Y))
issubgi.7	|- ((x e. Y /\ y e. Y) -> (xGy) e. Y)
issubgi.8	|- U e. Y
issubgi.9	|- (x e. Y -> (N` x) e. Y)

Assertion

Ref	Expression
issubgi	|- H e. (SubGrp` G)

Distinct variable groups:   x,H,y   y,N   x,U,y   x,Y,y

Proof of Theorem issubgi

Step	Hyp	Ref	Expression
1		issubgi.1	. 2 |- G e. Grp
2		issubgi.2	. . . . 5 |- X = ran G
3		rnexg 3359	. . . . . 6 |- (G e. Grp -> ran G e. V)
4	1, 3	ax-mp 7	. . . . 5 |- ran G e. V
5	2, 4	eqeltr 1544	. . . 4 |- X e. V
6		issubgi.5	. . . 4 |- Y (_ X
7	5, 6	ssexi 2720	. . 3 |- Y e. V
8		ffnoprval 4014	. . . 4 |- (H:(Y X. Y)-->Y <-> (H Fn (Y X. Y) /\ A.x e. Y A.y e. Y (xHy) e. Y))
9	2	grpfo 8043	. . . . . . . . 9 |- (G e. Grp -> G:(X X. X)-onto->X)
10		fof 3672	. . . . . . . . 9 |- (G:(X X. X)-onto->X -> G:(X X. X)-->X)
11	9, 10	syl 10	. . . . . . . 8 |- (G e. Grp -> G:(X X. X)-->X)
12	1, 11	ax-mp 7	. . . . . . 7 |- G:(X X. X)-->X
13		ssxp 3256	. . . . . . . 8 |- ((Y (_ X /\ Y (_ X) -> (Y X. Y) (_ (X X. X))
14	6, 6, 13	mp2an 697	. . . . . . 7 |- (Y X. Y) (_ (X X. X)
15		fssres 3643	. . . . . . 7 |- ((G:(X X. X)-->X /\ (Y X. Y) (_ (X X. X)) -> (G |` (Y X. Y)):(Y X. Y)-->X)
16	12, 14, 15	mp2an 697	. . . . . 6 |- (G |` (Y X. Y)):(Y X. Y)-->X
17		issubgi.6	. . . . . . 7 |- H = (G |` (Y X. Y))
18		feq1 3620	. . . . . . 7 |- (H = (G |` (Y X. Y)) -> (H:(Y X. Y)-->X <-> (G |` (Y X. Y)):(Y X. Y)-->X))
19	17, 18	ax-mp 7	. . . . . 6 |- (H:(Y X. Y)-->X <-> (G |` (Y X. Y)):(Y X. Y)-->X)
20	16, 19	mpbir 190	. . . . 5 |- H:(Y X. Y)-->X
21		ffn 3627	. . . . 5 |- (H:(Y X. Y)-->X -> H Fn (Y X. Y))
22	20, 21	ax-mp 7	. . . 4 |- H Fn (Y X. Y)
23		oprvalres 4033	. . . . . . . 8 |- ((a e. Y /\ b e. Y) -> (a(G |` (Y X. Y))b) = (aGb))
24	17	opreqi 3974	. . . . . . . 8 |- (aHb) = (a(G |` (Y X. Y))b)
25	23, 24	syl5eq 1519	. . . . . . 7 |- ((a e. Y /\ b e. Y) -> (aHb) = (aGb))
26	25	issubgilem 8121	. . . . . 6 |- ((x e. Y /\ y e. Y) -> (xHy) = (xGy))
27		issubgi.7	. . . . . 6 |- ((x e. Y /\ y e. Y) -> (xGy) e. Y)
28	26, 27	eqeltrd 1548	. . . . 5 |- ((x e. Y /\ y e. Y) -> (xHy) e. Y)
29	28	rgen2a 1699	. . . 4 |- A.x e. Y A.y e. Y (xHy) e. Y
30	8, 22, 29	mpbir2an 730	. . 3 |- H:(Y X. Y)-->Y
31	2	grpass 8047	. . . . . 6 |- ((G e. Grp /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Gz) = (xG(yGz)))
32	1, 31	mpan 695	. . . . 5 |- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))
33	6	sseli 2065	. . . . 5 |- (x e. Y -> x e. X)
34	6	sseli 2065	. . . . 5 |- (y e. Y -> y e. X)
35	6	sseli 2065	. . . . 5 |- (z e. Y -> z e. X)
36	32, 33, 34, 35	syl3an 868	. . . 4 |- ((x e. Y /\ y e. Y /\ z e. Y) -> ((xGy)Gz) = (xG(yGz)))
37	25	issubgilem 8121	. . . . . . 7 |- (((xHy) e. Y /\ z e. Y) -> ((xHy)Hz) = ((xHy)Gz))
38	37, 28	sylan 448	. . . . . 6 |- (((x e. Y /\ y e. Y) /\ z e. Y) -> ((xHy)Hz) = ((xHy)Gz))
39	38	3impa 828	. . . . 5 |- ((x e. Y /\ y e. Y /\ z e. Y) -> ((xHy)Hz) = ((xHy)Gz))
40	26	opreq1d 3975	. . . . . 6 |- ((x e. Y /\ y e. Y) -> ((xHy)Gz) = ((xGy)Gz))
41	40	3adant3 799	. . . . 5 |- ((x e. Y /\ y e. Y /\ z e. Y) -> ((xHy)Gz) = ((xGy)Gz))
42	39, 41	eqtrd 1507	. . . 4 |- ((x e. Y /\ y e. Y /\ z e. Y) -> ((xHy)Hz) = ((xGy)Gz))
43	25	issubgilem 8121	. . . . . . 7 |- ((x e. Y /\ (yHz) e. Y) -> (xH(yHz)) = (xG(yHz)))
44	30	foprcl 4015	. . . . . . 7 |- ((y e. Y /\ z e. Y) -> (yHz) e. Y)
45	43, 44	sylan2 451	. . . . . 6 |- ((x e. Y /\ (y e. Y /\ z e. Y)) -> (xH(yHz)) = (xG(yHz)))
46	45	3impb 829	. . . . 5 |- ((x e. Y /\ y e. Y /\ z e. Y) -> (xH(yHz)) = (xG(yHz)))
47	25	issubgilem 8121	. . . . . . 7 |- ((y e. Y /\ z e. Y) -> (yHz) = (yGz))
48	47	opreq2d 3976	. . . . . 6 |- ((y e. Y /\ z e. Y) -> (xG(yHz)) = (xG(yGz)))
49	48	3adant1 797	. . . . 5 |- ((x e. Y /\ y e. Y /\ z e. Y) -> (xG(yHz)) = (xG(yGz)))
50	46, 49	eqtrd 1507	. . . 4 |- ((x e. Y /\ y e. Y /\ z e. Y) -> (xH(yHz)) = (xG(yGz)))
51	36, 42, 50	3eqtr4d 1517	. . 3 |- ((x e. Y /\ y e. Y /\ z e. Y) -> ((xHy)Hz) = (xH(yHz)))
52		issubgi.8	. . 3 |- U e. Y
53	25	issubgilem 8121	. . . . 5 |- ((U e. Y /\ x e. Y) -> (UHx) = (UGx))
54	52, 53	mpan 695	. . . 4 |- (x e. Y -> (UHx) = (UGx))
55		issubgi.3	. . . . . . 7 |- U = (Id` G)
56	2, 55	grplid 8061	. . . . . 6 |- ((G e. Grp /\ x e. X) -> (UGx) = x)
57	1, 56	mpan 695	. . . . 5 |- (x e. X -> (UGx) = x)
58	33, 57	syl 10	. . . 4 |- (x e. Y -> (UGx) = x)
59	54, 58	eqtrd 1507	. . 3 |- (x e. Y -> (UHx) = x)
60		issubgi.9	. . 3 |- (x e. Y -> (N` x) e. Y)
61	25	issubgilem 8121	. . . . 5 |- (((N` x) e. Y /\ x e. Y) -> ((N` x)Hx) = ((N` x)Gx))
62	60, 61	mpancom 705	. . . 4 |- (x e. Y -> ((N` x)Hx) = ((N` x)Gx))
63		issubgi.4	. . . . . . 7 |- N = (inv` G)
64	2, 55, 63	grplinv 8070	. . . . . 6 |- ((G e. Grp /\ x e. X) -> ((N` x)Gx) = U)
65	1, 64	mpan 695	. . . . 5 |- (x e. X -> ((N` x)Gx) = U)
66	33, 65	syl 10	. . . 4 |- (x e. Y -> ((N` x)Gx) = U)
67	62, 66	eqtrd 1507	. . 3 |- (x e. Y -> ((N` x)Hx) = U)
68	7, 30, 51, 52, 59, 60, 67	isgrpi 8042	. 2 |- H e. Grp
69		resss 3383	. . 3 |- (G |` (Y X. Y)) (_ G
70	17, 69	eqsstr 2091	. 2 |- H (_ G
71		issubg 8116	. . 3 |- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H (_ G))
72	71	biimpr 152	. 2 |- ((G e. Grp /\ H e. Grp /\ H (_ G) -> H e. (SubGrp` G))
73	1, 68, 70, 72	mp3an 916	1 |- H e. (SubGrp` G)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   (_ wss 2047   X. cxp 3168  ran crn 3171   |` cres 3172   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034  invcgn 8035  SubGrpcsubg 8114

This theorem is referenced by:  readdsubg 8129  zaddsubg 8130  hhssabl 9132  ghomgrpilem2 10386

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037  df-gid 8038  df-ginv 8039  df-subg 8115

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