Theorem ghgrpi 9445

Description: The image of a group G under a group homomorphism F is a group, and furthermore is Abelian if G is Abelian. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator O in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.)

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))

Assertion

Ref	Expression
ghgrpi	|- (H e. Grp /\ (G e. Abel -> H e. Abel))

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

Proof of Theorem ghgrpi

Step	Hyp	Ref	Expression
1		ghgrpi.1	. . 3 |- G e. Grp
2		ghgrpi.2	. . 3 |- X = ran G
3		ghgrpi.3	. . 3 |- F:X-onto->Y
4		ghgrpi.4	. . 3 |- Y C_ A
5		ghgrpi.5	. . 3 |- O Fn (A X. A)
6		ghgrpi.6	. . 3 |- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))
7		ghgrpi.7	. . 3 |- H = (O |` (Y X. Y))
8		eqid 1884	. . 3 |- (Id` G) = (Id` G)
9		eqid 1884	. . 3 |- (inv` G) = (inv` G)
10		eqid 1884	. . 3 |- ( /g ` G) = ( /g ` G)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ghgrpilem4 9444	. 2 |- H e. Grp
12		fndm 4512	. . . . . . 7 |- (O Fn (A X. A) -> dom O = (A X. A))
13	5, 12	ax-mp 7	. . . . . 6 |- dom O = (A X. A)
14	7	resgrprn 9403	. . . . . 6 |- ((dom O = (A X. A) /\ H e. Grp /\ Y C_ A) -> Y = ran H)
15	13, 11, 4, 14	mp3an 1191	. . . . 5 |- Y = ran H
16	15	isabl 9409	. . . 4 |- (H e. Abel <-> (H e. Grp /\ A.a e. Y A.b e. Y (aHb) = (bHa)))
17	16	biimpri 169	. . 3 |- ((H e. Grp /\ A.a e. Y A.b e. Y (aHb) = (bHa)) -> H e. Abel)
18	2	ablcom 9411	. . . . . . . . . . . 12 |- ((G e. Abel /\ x e. X /\ y e. X) -> (xGy) = (yGx))
19	18	fveq2d 4685	. . . . . . . . . . 11 |- ((G e. Abel /\ x e. X /\ y e. X) -> (F` (xGy)) = (F` (yGx)))
20	1, 2, 3, 4, 5, 6, 7	ghgrpilem1 9441	. . . . . . . . . . . 12 |- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)H(F` y)))
21	20	3adant1 894	. . . . . . . . . . 11 |- ((G e. Abel /\ x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)H(F` y)))
22	1, 2, 3, 4, 5, 6, 7	ghgrpilem1 9441	. . . . . . . . . . . . 13 |- ((y e. X /\ x e. X) -> (F` (yGx)) = ((F` y)H(F` x)))
23	22	ancoms 484	. . . . . . . . . . . 12 |- ((x e. X /\ y e. X) -> (F` (yGx)) = ((F` y)H(F` x)))
24	23	3adant1 894	. . . . . . . . . . 11 |- ((G e. Abel /\ x e. X /\ y e. X) -> (F` (yGx)) = ((F` y)H(F` x)))
25	19, 21, 24	3eqtr3d 1934	. . . . . . . . . 10 |- ((G e. Abel /\ x e. X /\ y e. X) -> ((F` x)H(F` y)) = ((F` y)H(F` x)))
26	25	3coml 1075	. . . . . . . . 9 |- ((x e. X /\ y e. X /\ G e. Abel) -> ((F` x)H(F` y)) = ((F` y)H(F` x)))
27	26	3expb 1068	. . . . . . . 8 |- ((x e. X /\ (y e. X /\ G e. Abel)) -> ((F` x)H(F` y)) = ((F` y)H(F` x)))
28		opreq1 4889	. . . . . . . . 9 |- ((F` x) = a -> ((F` x)H(F` y)) = (aH(F` y)))
29		opreq2 4890	. . . . . . . . 9 |- ((F` x) = a -> ((F` y)H(F` x)) = ((F` y)Ha))
30	28, 29	eqeq12d 1899	. . . . . . . 8 |- ((F` x) = a -> (((F` x)H(F` y)) = ((F` y)H(F` x)) <-> (aH(F` y)) = ((F` y)Ha)))
31	1, 2, 3, 4, 5, 6, 7, 27, 30	ghgrpilem2 9442	. . . . . . 7 |- (((y e. X /\ G e. Abel) /\ a e. Y) -> (aH(F` y)) = ((F` y)Ha))
32	31	anasss 488	. . . . . 6 |- ((y e. X /\ (G e. Abel /\ a e. Y)) -> (aH(F` y)) = ((F` y)Ha))
33		opreq2 4890	. . . . . . 7 |- ((F` y) = b -> (aH(F` y)) = (aHb))
34		opreq1 4889	. . . . . . 7 |- ((F` y) = b -> ((F` y)Ha) = (bHa))
35	33, 34	eqeq12d 1899	. . . . . 6 |- ((F` y) = b -> ((aH(F` y)) = ((F` y)Ha) <-> (aHb) = (bHa)))
36	1, 2, 3, 4, 5, 6, 7, 32, 35	ghgrpilem2 9442	. . . . 5 |- (((G e. Abel /\ a e. Y) /\ b e. Y) -> (aHb) = (bHa))
37	36	expl 420	. . . 4 |- (G e. Abel -> ((a e. Y /\ b e. Y) -> (aHb) = (bHa)))
38	37	r19.21aivv 2183	. . 3 |- (G e. Abel -> A.a e. Y A.b e. Y (aHb) = (bHa))
39	17, 11, 38	sylancr 526	. 2 |- (G e. Abel -> H e. Abel)
40	11, 39	pm3.2i 307	1 |- (H e. Grp /\ (G e. Abel -> H e. Abel))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988   Fn wfn 3993  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  invcgn 9313   /g cgs 9314  Abelcabl 9407

This theorem is referenced by:  ghsubgi 9446

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  df-abl 9408

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