Theorem ghsubgi 8138

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 (_ 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 8116	. . . 4 |- (S e. (SubGrp` G) <-> (G e. Grp /\ S e. Grp /\ S (_ G))
3	1, 2	mpbi 189	. . 3 |- (G e. Grp /\ S e. Grp /\ S (_ G)
4	3	3simp2i 792	. 2 |- S e. Grp
5		ghsubgi.7	. 2 |- Z = ran S
6		ghsubgi.3	. . . . 5 |- F:X-->Y
7		ffun 3629	. . . . 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 8119	. . . . . 6 |- (S e. (SubGrp` G) -> Z (_ X)
11	1, 10	ax-mp 7	. . . . 5 |- Z (_ X
12	6	fdmi 3632	. . . . 5 |- dom F = X
13	11, 12	sseqtr4 2094	. . . 4 |- Z (_ dom F
14		fores 3681	. . . 4 |- ((Fun F /\ Z (_ dom F) -> (F |` Z):Z-onto->(F"Z))
15	8, 13, 14	mp2an 697	. . 3 |- (F |` Z):Z-onto->(F"Z)
16		ghsubgi.8	. . . 4 |- W = (F"Z)
17		foeq3 3670	. . . 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 190	. 2 |- (F |` Z):Z-onto->W
20		df-ima 3191	. . . . 5 |- (F"Z) = ran ( F |` Z)
21		fssres 3643	. . . . . . 7 |- ((F:X-->Y /\ Z (_ X) -> (F |` Z):Z-->Y)
22		frn 3633	. . . . . . 7 |- ((F |` Z):Z-->Y -> ran ( F |` Z) (_ Y)
23	21, 22	syl 10	. . . . . 6 |- ((F:X-->Y /\ Z (_ X) -> ran ( F |` Z) (_ Y)
24	6, 11, 23	mp2an 697	. . . . 5 |- ran ( F |` Z) (_ Y
25	20, 24	eqsstr 2091	. . . 4 |- (F"Z) (_ Y
26	16, 25	eqsstr 2091	. . 3 |- W (_ Y
27		ghsubgi.4	. . 3 |- Y (_ A
28	26, 27	sstri 2073	. 2 |- W (_ A
29		ghsubgi.5	. 2 |- O Fn (A X. A)
30	5	grpcl 8044	. . . . . 6 |- ((S e. Grp /\ x e. Z /\ y e. Z) -> (xSy) e. Z)
31	4, 30	mp3an1 903	. . . . 5 |- ((x e. Z /\ y e. Z) -> (xSy) e. Z)
32		fvres 3734	. . . . 5 |- ((xSy) e. Z -> ((F |` Z)` (xSy)) = (F` (xSy)))
33	31, 32	syl 10	. . . 4 |- ((x e. Z /\ y e. Z) -> ((F |` Z)` (xSy)) = (F` (xSy)))
34	5	subgopr 8118	. . . . . 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 3728	. . . 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 2065	. . . . 5 |- (x e. Z -> x e. X)
39	11	sseli 2065	. . . . 5 |- (y e. Z -> y e. X)
40	37, 38, 39	syl2an 454	. . . 4 |- ((x e. Z /\ y e. Z) -> (F` (xGy)) = ((F` x)O(F` y)))
41	33, 36, 40	3eqtrd 1511	. . 3 |- ((x e. Z /\ y e. Z) -> ((F |` Z)` (xSy)) = ((F` x)O(F` y)))
42		fvres 3734	. . . 4 |- (x e. Z -> ((F |` Z)` x) = (F` x))
43		fvres 3734	. . . 4 |- (y e. Z -> ((F |` Z)` y) = (F` y))
44	42, 43	opreqan12d 3979	. . 3 |- ((x e. Z /\ y e. Z) -> (((F |` Z)` x)O((F |` Z)` y)) = ((F` x)O(F` y)))
45	41, 44	eqtr4d 1510	. 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 8137	1 |- (H e. Grp /\ (S e. Abel -> H e. Abel))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   (_ wss 2047   X. cxp 3168  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173  Fun wfun 3176   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Abelcabl 8099  SubGrpcsubg 8114

This theorem is referenced by:  efghgrpilem 8719

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-grp 8037  df-gid 8038  df-ginv 8039  df-gdiv 8040  df-abl 8100  df-subg 8115

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