Theorem ghgrpi 8089

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 (_ 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 (_ 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 1473	. . 3 |- (Id` G) = (Id` G)
9		eqid 1473	. . 3 |- (inv` G) = (inv` G)
10		eqid 1473	. . 3 |- ( /g ` G) = ( /g ` G)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ghgrpilem4 8088	. 2 |- H e. Grp
12	2	ablcom 8054	. . . . . . . . . . . . 13 |- ((G e. Abel /\ x e. X /\ y e. X) -> (xGy) = (yGx))
13	12	fveq2d 3719	. . . . . . . . . . . 12 |- ((G e. Abel /\ x e. X /\ y e. X) -> (F` (xGy)) = (F` (yGx)))
14	1, 2, 3, 4, 5, 6, 7	ghgrpilem1 8085	. . . . . . . . . . . . 13 |- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)H(F` y)))
15	14	3adant1 796	. . . . . . . . . . . 12 |- ((G e. Abel /\ x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)H(F` y)))
16	1, 2, 3, 4, 5, 6, 7	ghgrpilem1 8085	. . . . . . . . . . . . . 14 |- ((y e. X /\ x e. X) -> (F` (yGx)) = ((F` y)H(F` x)))
17	16	ancoms 436	. . . . . . . . . . . . 13 |- ((x e. X /\ y e. X) -> (F` (yGx)) = ((F` y)H(F` x)))
18	17	3adant1 796	. . . . . . . . . . . 12 |- ((G e. Abel /\ x e. X /\ y e. X) -> (F` (yGx)) = ((F` y)H(F` x)))
19	13, 15, 18	3eqtr3d 1512	. . . . . . . . . . 11 |- ((G e. Abel /\ x e. X /\ y e. X) -> ((F` x)H(F` y)) = ((F` y)H(F` x)))
20	19	3coml 839	. . . . . . . . . 10 |- ((x e. X /\ y e. X /\ G e. Abel) -> ((F` x)H(F` y)) = ((F` y)H(F` x)))
21	20	3expb 833	. . . . . . . . 9 |- ((x e. X /\ (y e. X /\ G e. Abel)) -> ((F` x)H(F` y)) = ((F` y)H(F` x)))
22		opreq1 3959	. . . . . . . . . 10 |- ((F` x) = a -> ((F` x)H(F` y)) = (aH(F` y)))
23		opreq2 3960	. . . . . . . . . 10 |- ((F` x) = a -> ((F` y)H(F` x)) = ((F` y)Ha))
24	22, 23	eqeq12d 1486	. . . . . . . . 9 |- ((F` x) = a -> (((F` x)H(F` y)) = ((F` y)H(F` x)) <-> (aH(F` y)) = ((F` y)Ha)))
25	1, 2, 3, 4, 5, 6, 7, 21, 24	ghgrpilem2 8086	. . . . . . . 8 |- (((y e. X /\ G e. Abel) /\ a e. Y) -> (aH(F` y)) = ((F` y)Ha))
26	25	anasss 440	. . . . . . 7 |- ((y e. X /\ (G e. Abel /\ a e. Y)) -> (aH(F` y)) = ((F` y)Ha))
27		opreq2 3960	. . . . . . . 8 |- ((F` y) = b -> (aH(F` y)) = (aHb))
28		opreq1 3959	. . . . . . . 8 |- ((F` y) = b -> ((F` y)Ha) = (bHa))
29	27, 28	eqeq12d 1486	. . . . . . 7 |- ((F` y) = b -> ((aH(F` y)) = ((F` y)Ha) <-> (aHb) = (bHa)))
30	1, 2, 3, 4, 5, 6, 7, 26, 29	ghgrpilem2 8086	. . . . . 6 |- (((G e. Abel /\ a e. Y) /\ b e. Y) -> (aHb) = (bHa))
31	30	3impa 827	. . . . 5 |- ((G e. Abel /\ a e. Y /\ b e. Y) -> (aHb) = (bHa))
32	31	3expib 835	. . . 4 |- (G e. Abel -> ((a e. Y /\ b e. Y) -> (aHb) = (bHa)))
33	32	r19.21aivv 1717	. . 3 |- (G e. Abel -> A.a e. Y A.b e. Y (aHb) = (bHa))
34		fndm 3579	. . . . . . . 8 |- (O Fn (A X. A) -> dom O = (A X. A))
35	5, 34	ax-mp 7	. . . . . . 7 |- dom O = (A X. A)
36	7	resgrprn 8045	. . . . . . 7 |- ((dom O = (A X. A) /\ H e. Grp /\ Y (_ A) -> Y = ran H)
37	35, 11, 4, 36	mp3an 914	. . . . . 6 |- Y = ran H
38	37	isabl 8052	. . . . 5 |- (H e. Abel <-> (H e. Grp /\ A.a e. Y A.b e. Y (aHb) = (bHa)))
39	38	biimpr 152	. . . 4 |- ((H e. Grp /\ A.a e. Y A.b e. Y (aHb) = (bHa)) -> H e. Abel)
40	11, 39	mpan 694	. . 3 |- (A.a e. Y A.b e. Y (aHb) = (bHa) -> H e. Abel)
41	33, 40	syl 10	. 2 |- (G e. Abel -> H e. Abel)
42	11, 41	pm3.2i 285	1 |- (H e. Grp /\ (G e. Abel -> H e. Abel))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  A.wral 1642   (_ wss 2043   X. cxp 3163  dom cdm 3165  ran crn 3166   |` cres 3167   Fn wfn 3172  -onto->wfo 3175  ` cfv 3177  (class class class)co 3954  Grpcgr 7983  Idcgi 7984  invcgn 7985   /g cgs 7986  Abelcabl 8050

This theorem is referenced by:  ghsubgi 8090

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-grp 7987  df-gid 7988  df-ginv 7989  df-gdiv 7990  df-abl 8051

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