Theorem ghsubgi 9446

Description: The image of a subgroup S of group G under a group homomorphism F on G is a group, and furthermore is Abelian if S is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.)

Hypotheses

Ref	Expression
ghsubgi.1	|- S e. (SubGrp` G)
ghsubgi.2	|- X = ran G
ghsubgi.3	|- F:X-->Y
ghsubgi.4	|- Y C_ A
ghsubgi.5	|- O Fn (A X. A)
ghsubgi.6	|- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))
ghsubgi.7	|- Z = ran S
ghsubgi.8	|- W = (F"Z)
ghsubgi.9	|- H = (O |` (W X. W))

Assertion

Ref	Expression
ghsubgi	|- (H e. Grp /\ (S e. Abel -> H e. Abel))

Distinct variable groups:   x,F,y   x,H,y   x,O,y   x,S,y   x,W,y   x,Z,y

Proof of Theorem ghsubgi

Step	Hyp	Ref	Expression
1		ghsubgi.1	. . . 4 |- S e. (SubGrp` G)
2		issubg 9425	. . . 4 |- (S e. (SubGrp` G) <-> (G e. Grp /\ S e. Grp /\ S C_ G))
3	1, 2	mpbi 206	. . 3 |- (G e. Grp /\ S e. Grp /\ S C_ G)
4	3	simp2i 886	. 2 |- S e. Grp
5		ghsubgi.7	. 2 |- Z = ran S
6		ghsubgi.3	. . . . 5 |- F:X-->Y
7		ffun 4565	. . . . 5 |- (F:X-->Y -> Fun F)
8	6, 7	ax-mp 7	. . . 4 |- Fun F
9		ghsubgi.2	. . . . . . 7 |- X = ran G
10	9, 5	subgrnss 9428	. . . . . 6 |- (S e. (SubGrp` G) -> Z C_ X)
11	1, 10	ax-mp 7	. . . . 5 |- Z C_ X
12	6	fdmi 4568	. . . . 5 |- dom F = X
13	11, 12	sseqtr4i 2650	. . . 4 |- Z C_ dom F
14		fores 4627	. . . 4 |- ((Fun F /\ Z C_ dom F) -> (F |` Z):Z-onto->(F"Z))
15	8, 13, 14	mp2an 761	. . 3 |- (F |` Z):Z-onto->(F"Z)
16		ghsubgi.8	. . . 4 |- W = (F"Z)
17		foeq3 4615	. . . 4 |- (W = (F"Z) -> ((F |` Z):Z-onto->W <-> (F |` Z):Z-onto->(F"Z)))
18	16, 17	ax-mp 7	. . 3 |- ((F |` Z):Z-onto->W <-> (F |` Z):Z-onto->(F"Z))
19	15, 18	mpbir 207	. 2 |- (F |` Z):Z-onto->W
20		df-ima 4007	. . . . 5 |- (F"Z) = ran ( F |` Z)
21		fssres 4582	. . . . . . 7 |- ((F:X-->Y /\ Z C_ X) -> (F |` Z):Z-->Y)
22		frn 4569	. . . . . . 7 |- ((F |` Z):Z-->Y -> ran ( F |` Z) C_ Y)
23	21, 22	syl 12	. . . . . 6 |- ((F:X-->Y /\ Z C_ X) -> ran ( F |` Z) C_ Y)
24	6, 11, 23	mp2an 761	. . . . 5 |- ran ( F |` Z) C_ Y
25	20, 24	eqsstri 2647	. . . 4 |- (F"Z) C_ Y
26	16, 25	eqsstri 2647	. . 3 |- W C_ Y
27		ghsubgi.4	. . 3 |- Y C_ A
28	26, 27	sstri 2626	. 2 |- W C_ A
29		ghsubgi.5	. 2 |- O Fn (A X. A)
30	5	grpcl 9324	. . . . . 6 |- ((S e. Grp /\ x e. Z /\ y e. Z) -> (xSy) e. Z)
31	4, 30	mp3an1 1178	. . . . 5 |- ((x e. Z /\ y e. Z) -> (xSy) e. Z)
32		fvres 4691	. . . . 5 |- ((xSy) e. Z -> ((F |` Z)` (xSy)) = (F` (xSy)))
33	31, 32	syl 12	. . . 4 |- ((x e. Z /\ y e. Z) -> ((F |` Z)` (xSy)) = (F` (xSy)))
34	5	subgopr 9427	. . . . . 6 |- (S e. (SubGrp` G) -> ((x e. Z /\ y e. Z) -> (xSy) = (xGy)))
35	1, 34	ax-mp 7	. . . . 5 |- ((x e. Z /\ y e. Z) -> (xSy) = (xGy))
36	35	fveq2d 4685	. . . 4 |- ((x e. Z /\ y e. Z) -> (F` (xSy)) = (F` (xGy)))
37		ghsubgi.6	. . . . 5 |- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))
38	11	sseli 2617	. . . . 5 |- (x e. Z -> x e. X)
39	11	sseli 2617	. . . . 5 |- (y e. Z -> y e. X)
40	37, 38, 39	syl2an 503	. . . 4 |- ((x e. Z /\ y e. Z) -> (F` (xGy)) = ((F` x)O(F` y)))
41	33, 36, 40	3eqtrd 1929	. . 3 |- ((x e. Z /\ y e. Z) -> ((F |` Z)` (xSy)) = ((F` x)O(F` y)))
42		fvres 4691	. . . 4 |- (x e. Z -> ((F |` Z)` x) = (F` x))
43		fvres 4691	. . . 4 |- (y e. Z -> ((F |` Z)` y) = (F` y))
44	42, 43	opreqan12d 4902	. . 3 |- ((x e. Z /\ y e. Z) -> (((F |` Z)` x)O((F |` Z)` y)) = ((F` x)O(F` y)))
45	41, 44	eqtr4d 1928	. 2 |- ((x e. Z /\ y e. Z) -> ((F |` Z)` (xSy)) = (((F |` Z)` x)O((F |` Z)` y)))
46		ghsubgi.9	. 2 |- H = (O |` (W X. W))
47	4, 5, 19, 28, 29, 45, 46	ghgrpi 9445	1 |- (H e. Grp /\ (S e. Abel -> H e. Abel))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Abelcabl 9407  SubGrpcsubg 9423

This theorem is referenced by:  efghgrpilem 10073

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  df-subg 9424

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