Theorem ghgrpilem4 9444

Description: Lemma for ghgrpi 9445.

Hypotheses

Ref	Expression
ghgrpi.1	|- G e. Grp
ghgrpi.2	|- X = ran G
ghgrpi.3	|- F:X-onto->Y
ghgrpi.4	|- Y C_ A
ghgrpi.5	|- O Fn (A X. A)
ghgrpi.6	|- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))
ghgrpi.7	|- H = (O |` (Y X. Y))
ghgrpilem3.8	|- U = (Id` G)
ghgrpilem3.9	|- N = (inv` G)
ghgrpilem3.10	|- D = ( /g ` G)

Assertion

Ref	Expression
ghgrpilem4	|- H e. Grp

Distinct variable groups:   x,F,y   x,G,y   x,H,y   x,O,y   x,X,y   x,Y,y

Proof of Theorem ghgrpilem4

Step	Hyp	Ref	Expression
1		ghgrpi.2	. . . 4 |- X = ran G
2		ghgrpi.1	. . . . 5 |- G e. Grp
3		rnexg 4207	. . . . 5 |- (G e. Grp -> ran G e. _V)
4	2, 3	ax-mp 7	. . . 4 |- ran G e. _V
5	1, 4	eqeltri 1967	. . 3 |- X e. _V
6		ghgrpi.3	. . 3 |- F:X-onto->Y
7		fornex 4625	. . 3 |- (X e. _V -> (F:X-onto->Y -> Y e. _V))
8	5, 6, 7	mp2 54	. 2 |- Y e. _V
9		df-fo 4012	. . . . . . 7 |- (F:X-onto->Y <-> (F Fn X /\ ran F = Y))
10	6, 9	mpbi 206	. . . . . 6 |- (F Fn X /\ ran F = Y)
11	10	simpli 347	. . . . 5 |- F Fn X
12		ghgrpilem3.8	. . . . . . 7 |- U = (Id` G)
13	1, 12	grpidcl 9343	. . . . . 6 |- (G e. Grp -> U e. X)
14	2, 13	ax-mp 7	. . . . 5 |- U e. X
15		fnfvelrn 4786	. . . . 5 |- ((F Fn X /\ U e. X) -> (F` U) e. ran F)
16	11, 14, 15	mp2an 761	. . . 4 |- (F` U) e. ran F
17	10	simpri 351	. . . 4 |- ran F = Y
18	16, 17	eleqtri 1969	. . 3 |- (F` U) e. Y
19		ne0i 2881	. . 3 |- ((F` U) e. Y -> Y =/= (/))
20	18, 19	ax-mp 7	. 2 |- Y =/= (/)
21		ffnoprv 4943	. . 3 |- (H:(Y X. Y)-->Y <-> (H Fn (Y X. Y) /\ A.a e. Y A.b e. Y (aHb) e. Y))
22		ghgrpi.5	. . . . 5 |- O Fn (A X. A)
23		ghgrpi.4	. . . . . 6 |- Y C_ A
24		xpss12 4089	. . . . . 6 |- ((Y C_ A /\ Y C_ A) -> (Y X. Y) C_ (A X. A))
25	23, 23, 24	mp2an 761	. . . . 5 |- (Y X. Y) C_ (A X. A)
26		fnssres 4526	. . . . 5 |- ((O Fn (A X. A) /\ (Y X. Y) C_ (A X. A)) -> (O |` (Y X. Y)) Fn (Y X. Y))
27	22, 25, 26	mp2an 761	. . . 4 |- (O |` (Y X. Y)) Fn (Y X. Y)
28		ghgrpi.7	. . . . 5 |- H = (O |` (Y X. Y))
29	28	fneq1i 4507	. . . 4 |- (H Fn (Y X. Y) <-> (O |` (Y X. Y)) Fn (Y X. Y))
30	27, 29	mpbir 207	. . 3 |- H Fn (Y X. Y)
31		ghgrpi.6	. . . . 5 |- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))
32	2, 1, 6, 23, 22, 31, 28	ghgrpilem1 9441	. . . . . . . 8 |- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)H(F` y)))
33		fnfvelrn 4786	. . . . . . . . 9 |- ((F Fn X /\ (xGy) e. X) -> (F` (xGy)) e. ran F)
34	1	grpcl 9324	. . . . . . . . . 10 |- ((G e. Grp /\ x e. X /\ y e. X) -> (xGy) e. X)
35	2, 34	mp3an1 1178	. . . . . . . . 9 |- ((x e. X /\ y e. X) -> (xGy) e. X)
36	33, 11, 35	sylancr 526	. . . . . . . 8 |- ((x e. X /\ y e. X) -> (F` (xGy)) e. ran F)
37	32, 36	eqeltrrd 1972	. . . . . . 7 |- ((x e. X /\ y e. X) -> ((F` x)H(F` y)) e. ran F)
38	37, 17	syl6eleq 1981	. . . . . 6 |- ((x e. X /\ y e. X) -> ((F` x)H(F` y)) e. Y)
39		opreq1 4889	. . . . . . 7 |- ((F` x) = a -> ((F` x)H(F` y)) = (aH(F` y)))
40	39	eleq1d 1963	. . . . . 6 |- ((F` x) = a -> (((F` x)H(F` y)) e. Y <-> (aH(F` y)) e. Y))
41	2, 1, 6, 23, 22, 31, 28, 38, 40	ghgrpilem2 9442	. . . . 5 |- ((y e. X /\ a e. Y) -> (aH(F` y)) e. Y)
42		opreq2 4890	. . . . . 6 |- ((F` y) = b -> (aH(F` y)) = (aHb))
43	42	eleq1d 1963	. . . . 5 |- ((F` y) = b -> ((aH(F` y)) e. Y <-> (aHb) e. Y))
44	2, 1, 6, 23, 22, 31, 28, 41, 43	ghgrpilem2 9442	. . . 4 |- ((a e. Y /\ b e. Y) -> (aHb) e. Y)
45	44	rgen2a 2160	. . 3 |- A.a e. Y A.b e. Y (aHb) e. Y
46	21, 30, 45	mpbir2an 800	. 2 |- H:(Y X. Y)-->Y
47	1	grpass 9327	. . . . . . . . . . . . 13 |- ((G e. Grp /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Gz) = (xG(yGz)))
48	2, 47	mpan 759	. . . . . . . . . . . 12 |- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))
49	48	fveq2d 4685	. . . . . . . . . . 11 |- ((x e. X /\ y e. X /\ z e. X) -> (F` ((xGy)Gz)) = (F` (xG(yGz))))
50	2, 1, 6, 23, 22, 31, 28	ghgrpilem1 9441	. . . . . . . . . . . . 13 |- (((xGy) e. X /\ z e. X) -> (F` ((xGy)Gz)) = ((F` (xGy))H(F` z)))
51	50, 35	sylan 497	. . . . . . . . . . . 12 |- (((x e. X /\ y e. X) /\ z e. X) -> (F` ((xGy)Gz)) = ((F` (xGy))H(F` z)))
52	51	3impa 1062	. . . . . . . . . . 11 |- ((x e. X /\ y e. X /\ z e. X) -> (F` ((xGy)Gz)) = ((F` (xGy))H(F` z)))
53	2, 1, 6, 23, 22, 31, 28	ghgrpilem1 9441	. . . . . . . . . . . . 13 |- ((x e. X /\ (yGz) e. X) -> (F` (xG(yGz))) = ((F` x)H(F` (yGz))))
54	1	grpcl 9324	. . . . . . . . . . . . . 14 |- ((G e. Grp /\ y e. X /\ z e. X) -> (yGz) e. X)
55	2, 54	mp3an1 1178	. . . . . . . . . . . . 13 |- ((y e. X /\ z e. X) -> (yGz) e. X)
56	53, 55	sylan2 500	. . . . . . . . . . . 12 |- ((x e. X /\ (y e. X /\ z e. X)) -> (F` (xG(yGz))) = ((F` x)H(F` (yGz))))
57	56	3impb 1063	. . . . . . . . . . 11 |- ((x e. X /\ y e. X /\ z e. X) -> (F` (xG(yGz))) = ((F` x)H(F` (yGz))))
58	49, 52, 57	3eqtr3d 1934	. . . . . . . . . 10 |- ((x e. X /\ y e. X /\ z e. X) -> ((F` (xGy))H(F` z)) = ((F` x)H(F` (yGz))))
59	32	3adant3 896	. . . . . . . . . . 11 |- ((x e. X /\ y e. X /\ z e. X) -> (F` (xGy)) = ((F` x)H(F` y)))
60	59	opreq1d 4897	. . . . . . . . . 10 |- ((x e. X /\ y e. X /\ z e. X) -> ((F` (xGy))H(F` z)) = (((F` x)H(F` y))H(F` z)))
61	2, 1, 6, 23, 22, 31, 28	ghgrpilem1 9441	. . . . . . . . . . . 12 |- ((y e. X /\ z e. X) -> (F` (yGz)) = ((F` y)H(F` z)))
62	61	3adant1 894	. . . . . . . . . . 11 |- ((x e. X /\ y e. X /\ z e. X) -> (F` (yGz)) = ((F` y)H(F` z)))
63	62	opreq2d 4898	. . . . . . . . . 10 |- ((x e. X /\ y e. X /\ z e. X) -> ((F` x)H(F` (yGz))) = ((F` x)H((F` y)H(F` z))))
64	58, 60, 63	3eqtr3d 1934	. . . . . . . . 9 |- ((x e. X /\ y e. X /\ z e. X) -> (((F` x)H(F` y))H(F` z)) = ((F` x)H((F` y)H(F` z))))
65	64	3expb 1068	. . . . . . . 8 |- ((x e. X /\ (y e. X /\ z e. X)) -> (((F` x)H(F` y))H(F` z)) = ((F` x)H((F` y)H(F` z))))
66	39	opreq1d 4897	. . . . . . . . 9 |- ((F` x) = a -> (((F` x)H(F` y))H(F` z)) = ((aH(F` y))H(F` z)))
67		opreq1 4889	. . . . . . . . 9 |- ((F` x) = a -> ((F` x)H((F` y)H(F` z))) = (aH((F` y)H(F` z))))
68	66, 67	eqeq12d 1899	. . . . . . . 8 |- ((F` x) = a -> ((((F` x)H(F` y))H(F` z)) = ((F` x)H((F` y)H(F` z))) <-> ((aH(F` y))H(F` z)) = (aH((F` y)H(F` z)))))
69	2, 1, 6, 23, 22, 31, 28, 65, 68	ghgrpilem2 9442	. . . . . . 7 |- (((y e. X /\ z e. X) /\ a e. Y) -> ((aH(F` y))H(F` z)) = (aH((F` y)H(F` z))))
70	69	anasss 488	. . . . . 6 |- ((y e. X /\ (z e. X /\ a e. Y)) -> ((aH(F` y))H(F` z)) = (aH((F` y)H(F` z))))
71	42	opreq1d 4897	. . . . . . 7 |- ((F` y) = b -> ((aH(F` y))H(F` z)) = ((aHb)H(F` z)))
72		opreq1 4889	. . . . . . . 8 |- ((F` y) = b -> ((F` y)H(F` z)) = (bH(F` z)))
73	72	opreq2d 4898	. . . . . . 7 |- ((F` y) = b -> (aH((F` y)H(F` z))) = (aH(bH(F` z))))
74	71, 73	eqeq12d 1899	. . . . . 6 |- ((F` y) = b -> (((aH(F` y))H(F` z)) = (aH((F` y)H(F` z))) <-> ((aHb)H(F` z)) = (aH(bH(F` z)))))
75	2, 1, 6, 23, 22, 31, 28, 70, 74	ghgrpilem2 9442	. . . . 5 |- (((z e. X /\ a e. Y) /\ b e. Y) -> ((aHb)H(F` z)) = (aH(bH(F` z))))
76	75	anasss 488	. . . 4 |- ((z e. X /\ (a e. Y /\ b e. Y)) -> ((aHb)H(F` z)) = (aH(bH(F` z))))
77		opreq2 4890	. . . . 5 |- ((F` z) = c -> ((aHb)H(F` z)) = ((aHb)Hc))
78		opreq2 4890	. . . . . 6 |- ((F` z) = c -> (bH(F` z)) = (bHc))
79	78	opreq2d 4898	. . . . 5 |- ((F` z) = c -> (aH(bH(F` z))) = (aH(bHc)))
80	77, 79	eqeq12d 1899	. . . 4 |- ((F` z) = c -> (((aHb)H(F` z)) = (aH(bH(F` z))) <-> ((aHb)Hc) = (aH(bHc))))
81	2, 1, 6, 23, 22, 31, 28, 76, 80	ghgrpilem2 9442	. . 3 |- (((a e. Y /\ b e. Y) /\ c e. Y) -> ((aHb)Hc) = (aH(bHc)))
82	81	3impa 1062	. 2 |- ((a e. Y /\ b e. Y /\ c e. Y) -> ((aHb)Hc) = (aH(bHc)))
83		ghgrpilem3.9	. . . 4 |- N = (inv` G)
84		ghgrpilem3.10	. . . 4 |- D = ( /g ` G)
85	2, 1, 6, 23, 22, 31, 28, 12, 83, 84	ghgrpilem3 9443	. . 3 |- ((a e. Y /\ b e. Y) -> (E.c e. Y (cHa) = b /\ E.c e. Y (aHc) = b))
86	85	simplld 348	. 2 |- ((a e. Y /\ b e. Y) -> E.c e. Y (cHa) = b)
87	85	simprd 352	. 2 |- ((a e. Y /\ b e. Y) -> E.c e. Y (aHc) = b)
88	8, 20, 46, 82, 86, 87	isgrp2i 9360	1 |- H e. Grp

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  (/)c0 2875   X. cxp 3984  ran crn 3987   |` cres 3988   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  invcgn 9313   /g cgs 9314

This theorem is referenced by:  ghgrpi 9445

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-oprab 4887  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319

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