Theorem cayleylem3 10406

Description: Lemma for cayleyi 10407.

Hypotheses

Ref	Expression
cayleylem1.1	|- G e. Grp
cayleylem1.2	|- P = {f | f:X-1-1-onto->X}
cayleylem1.3	|- X = ran G
cayleylem1.4	|- U = (Id` G)
cayleylem1.5	|- H = (SymGrp` X)
cayleylem1.6	|- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
cayleylem1.7	|- Y = ran F
cayleylem1.8	|- S = (H |` (Y X. Y))

Assertion

Ref	Expression
cayleylem3	|- (S e. (SubGrp` H) /\ F e. (G GrpIso S))

Distinct variable groups:   a,b,f,g,h   f,F   G,a,b,f,g,h   P,g,h   U,a,b   X,a,b,f,g,h

Proof of Theorem cayleylem3

Step	Hyp	Ref	Expression
1		cayleylem1.1	. . 3 |- G e. Grp
2		cayleylem1.5	. . . 4 |- H = (SymGrp` X)
3		cayleylem1.3	. . . . . 6 |- X = ran G
4		rnexg 3365	. . . . . . 7 |- (G e. Grp -> ran G e. V)
5	1, 4	ax-mp 7	. . . . . 6 |- ran G e. V
6	3, 5	eqeltr 1547	. . . . 5 |- X e. V
7	6	symggrpi 10401	. . . 4 |- (SymGrp` X) e. Grp
8	2, 7	eqeltr 1547	. . 3 |- H e. Grp
9		cayleylem1.2	. . . 4 |- P = {f | f:X-1-1-onto->X}
10		cayleylem1.4	. . . 4 |- U = (Id` G)
11		cayleylem1.6	. . . 4 |- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
12		cayleylem1.7	. . . 4 |- Y = ran F
13		cayleylem1.8	. . . 4 |- S = (H |` (Y X. Y))
14	1, 9, 3, 10, 2, 11, 12, 13	cayleylem2 10405	. . 3 |- F e. (G GrpHom H)
15	1, 8, 14, 12, 13	ghomgrpi 10382	. 2 |- S e. (SubGrp` H)
16		issubg 8112	. . . . . 6 |- (S e. (SubGrp` H) <-> (H e. Grp /\ S e. Grp /\ S (_ H))
17	15, 16	mpbi 189	. . . . 5 |- (H e. Grp /\ S e. Grp /\ S (_ H)
18	17	3simp2i 794	. . . 4 |- S e. Grp
19		elgiso 10393	. . . 4 |- ((G e. Grp /\ S e. Grp) -> (F e. (G GrpIso S) <-> (F e. (G GrpHom S) /\ F:ran G-1-1-onto->ran S)))
20	1, 18, 19	mp2an 699	. . 3 |- (F e. (G GrpIso S) <-> (F e. (G GrpHom S) /\ F:ran G-1-1-onto->ran S))
21	12, 13	ghomgsg 10390	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F e. (G GrpHom S))
22	1, 8, 14, 21	mp3an 918	. . 3 |- F e. (G GrpHom S)
23		eqid 1478	. . . . . . . . . . . . . . . 16 |- ran S = ran S
24	3, 12, 13, 23	ghomfo 10386	. . . . . . . . . . . . . . 15 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-onto->ran S)
25		forn 3680	. . . . . . . . . . . . . . 15 |- (F:X-onto->ran S -> ran F = ran S)
26	24, 25	syl 10	. . . . . . . . . . . . . 14 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ran F = ran S)
27	1, 8, 14, 26	mp3an 918	. . . . . . . . . . . . 13 |- ran F = ran S
28	12, 27	eqtr 1498	. . . . . . . . . . . 12 |- Y = ran S
29	28	grpfo 8040	. . . . . . . . . . 11 |- (S e. Grp -> S:(Y X. Y)-onto->Y)
30		fof 3678	. . . . . . . . . . 11 |- (S:(Y X. Y)-onto->Y -> S:(Y X. Y)-->Y)
31		ffn 3633	. . . . . . . . . . 11 |- (S:(Y X. Y)-->Y -> S Fn (Y X. Y))
32	29, 30, 31	3syl 20	. . . . . . . . . 10 |- (S e. Grp -> S Fn (Y X. Y))
33		fnresdm 3602	. . . . . . . . . 10 |- (S Fn (Y X. Y) -> (S |` (Y X. Y)) = S)
34	32, 33	syl 10	. . . . . . . . 9 |- (S e. Grp -> (S |` (Y X. Y)) = S)
35	18, 34	ax-mp 7	. . . . . . . 8 |- (S |` (Y X. Y)) = S
36	35	eqcomi 1482	. . . . . . 7 |- S = (S |` (Y X. Y))
37		eqid 1478	. . . . . . 7 |- (Id` S) = (Id` S)
38	3, 12, 36, 28, 10, 37	ghomf1o 10392	. . . . . 6 |- ((G e. Grp /\ S e. Grp /\ F e. (G GrpHom S)) -> (F:X-1-1-onto->Y <-> A.y e. X ((F` y) = (Id` S) -> y = U)))
39	1, 18, 22, 38	mp3an 918	. . . . 5 |- (F:X-1-1-onto->Y <-> A.y e. X ((F` y) = (Id` S) -> y = U))
40	3, 10	grpidcl 8055	. . . . . . . . . 10 |- (G e. Grp -> U e. X)
41	1, 40	ax-mp 7	. . . . . . . . 9 |- U e. X
42	11, 3	grplactval 8093	. . . . . . . . 9 |- ((G e. Grp /\ y e. X /\ U e. X) -> ((F` y)` U) = (yGU))
43	1, 41, 42	mp3an13 909	. . . . . . . 8 |- (y e. X -> ((F` y)` U) = (yGU))
44	3, 10	grprid 8058	. . . . . . . . 9 |- ((G e. Grp /\ y e. X) -> (yGU) = y)
45	1, 44	mpan 697	. . . . . . . 8 |- (y e. X -> (yGU) = y)
46	43, 45	eqtrd 1510	. . . . . . 7 |- (y e. X -> ((F` y)` U) = y)
47	46	eqeq1d 1486	. . . . . 6 |- (y e. X -> (((F` y)` U) = U <-> y = U))
48		eqid 1478	. . . . . . . . . . 11 |- (Id` H) = (Id` H)
49	48, 37	subgid 8116	. . . . . . . . . 10 |- (S e. (SubGrp` H) -> (Id` S) = (Id` H))
50	15, 49	ax-mp 7	. . . . . . . . 9 |- (Id` S) = (Id` H)
51	2	fveq2i 3733	. . . . . . . . . 10 |- (Id` H) = (Id` (SymGrp` X))
52	6	symgidi 10402	. . . . . . . . . 10 |- (Id` (SymGrp` X)) = (I |` X)
53	51, 52	eqtr 1498	. . . . . . . . 9 |- (Id` H) = (I |` X)
54	50, 53	eqtr2 1499	. . . . . . . 8 |- (I |` X) = (Id` S)
55	54	eqeq2i 1488	. . . . . . 7 |- ((F` y) = (I |` X) <-> (F` y) = (Id` S))
56		fveq1 3729	. . . . . . . 8 |- ((F` y) = (I |` X) -> ((F` y)` U) = ((I |` X)` U))
57		fvresi 3849	. . . . . . . . 9 |- (U e. X -> ((I |` X)` U) = U)
58	41, 57	ax-mp 7	. . . . . . . 8 |- ((I |` X)` U) = U
59	56, 58	syl6eq 1526	. . . . . . 7 |- ((F` y) = (I |` X) -> ((F` y)` U) = U)
60	55, 59	sylbir 201	. . . . . 6 |- ((F` y) = (Id` S) -> ((F` y)` U) = U)
61	47, 60	syl5bi 208	. . . . 5 |- (y e. X -> ((F` y) = (Id` S) -> y = U))
62	39, 61	mprgbir 1704	. . . 4 |- F:X-1-1-onto->Y
63		f1oeq3 3692	. . . . . 6 |- (Y = ran S -> (F:X-1-1-onto->Y <-> F:X-1-1-onto->ran S))
64	28, 63	ax-mp 7	. . . . 5 |- (F:X-1-1-onto->Y <-> F:X-1-1-onto->ran S)
65		f1oeq2 3691	. . . . . 6 |- (X = ran G -> (F:X-1-1-onto->ran S <-> F:ran G-1-1-onto->ran S))
66	3, 65	ax-mp 7	. . . . 5 |- (F:X-1-1-onto->ran S <-> F:ran G-1-1-onto->ran S)
67	64, 66	bitr 173	. . . 4 |- (F:X-1-1-onto->Y <-> F:ran G-1-1-onto->ran S)
68	62, 67	mpbi 189	. . 3 |- F:ran G-1-1-onto->ran S
69	20, 22, 68	mpbir2an 732	. 2 |- F e. (G GrpIso S)
70	15, 69	pm3.2i 285	1 |- (S e. (SubGrp` H) /\ F e. (G GrpIso S))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  {cab 1466  A.wral 1648  Vcvv 1814   (_ wss 2050  {copab 2671  Icid 2837   X. cxp 3174  ran crn 3177   |` cres 3178   Fn wfn 3183  -->wf 3184  -onto->wfo 3186  -1-1-onto->wf1o 3187  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031  SubGrpcsubg 8110   GrpHom cghom 10373   GrpIso cgiso 10374  SymGrpcsymgrp 10394

This theorem is referenced by:  cayleyi 10407

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-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  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-oprab 3972  df-1st 4085  df-2nd 4086  df-grp 8034  df-gid 8035  df-ginv 8036  df-subg 8111  df-ghom 10375  df-giso 10376  df-symgrp 10395

Copyright terms: Public domain