Theorem ghsubgolemOLD 26179

Description: Obsolete as of 14-Mar-2020. 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.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
ghsubgolemOLD	|- ( ph -> ( H e. GrpOp /\ ( S e. AbelOp -> H e. AbelOp ) ) )

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

Allowed substitution hints:   A( x, y)   G( x, y)   X( x, y)   Y( x, y)

Proof of Theorem ghsubgolemOLD

Step	Hyp	Ref	Expression
1		ghsubgoOLD.3	. . . . 5 |- ( ph -> F : X --> Y )
2		ffun 5742	. . . . 5 |- ( F : X --> Y -> Fun F )
3	1, 2	syl 17	. . . 4 |- ( ph -> Fun F )
4		ghsubgoOLD.1	. . . . . 6 |- ( ph -> S e. ( SubGrpOp ` G ) )
5		ghsubgoOLD.2	. . . . . . 7 |- X = ran G
6		ghsubgoOLD.7	. . . . . . 7 |- Z = ran S
7	5, 6	subgornss 26115	. . . . . 6 |- ( S e. ( SubGrpOp ` G ) -> Z C_ X )
8	4, 7	syl 17	. . . . 5 |- ( ph -> Z C_ X )
9		fdm 5745	. . . . . 6 |- ( F : X --> Y -> dom F = X )
10	1, 9	syl 17	. . . . 5 |- ( ph -> dom F = X )
11	8, 10	sseqtr4d 3455	. . . 4 |- ( ph -> Z C_ dom F )
12		fores 5815	. . . 4 |- ( ( Fun F /\ Z C_ dom F ) -> ( F |` Z ) : Z -onto-> ( F " Z ) )
13	3, 11, 12	syl2anc 673	. . 3 |- ( ph -> ( F |` Z ) : Z -onto-> ( F " Z ) )
14		ssel2 3413	. . . . . . 7 |- ( ( Z C_ X /\ x e. Z ) -> x e. X )
15		ssel2 3413	. . . . . . 7 |- ( ( Z C_ X /\ y e. Z ) -> y e. X )
16	14, 15	anim12dan 855	. . . . . 6 |- ( ( Z C_ X /\ ( x e. Z /\ y e. Z ) ) -> ( x e. X /\ y e. X ) )
17	8, 16	sylan 479	. . . . 5 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( x e. X /\ y e. X ) )
18		ghsubgoOLD.6	. . . . 5 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( F ` ( x G y ) ) = ( ( F ` x ) O ( F ` y ) ) )
19	17, 18	syldan 478	. . . 4 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( F ` ( x G y ) ) = ( ( F ` x ) O ( F ` y ) ) )
20		issubgo 26112	. . . . . . . . 9 |- ( S e. ( SubGrpOp ` G ) <-> ( G e. GrpOp /\ S e. GrpOp /\ S C_ G ) )
21	20	simp2bi 1046	. . . . . . . 8 |- ( S e. ( SubGrpOp ` G ) -> S e. GrpOp )
22	4, 21	syl 17	. . . . . . 7 |- ( ph -> S e. GrpOp )
23	6	grpocl 26009	. . . . . . . 8 |- ( ( S e. GrpOp /\ x e. Z /\ y e. Z ) -> ( x S y ) e. Z )
24	23	3expb 1232	. . . . . . 7 |- ( ( S e. GrpOp /\ ( x e. Z /\ y e. Z ) ) -> ( x S y ) e. Z )
25	22, 24	sylan 479	. . . . . 6 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( x S y ) e. Z )
26		fvres 5893	. . . . . 6 |- ( ( x S y ) e. Z -> ( ( F |` Z ) ` ( x S y ) ) = ( F ` ( x S y ) ) )
27	25, 26	syl 17	. . . . 5 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( ( F |` Z ) ` ( x S y ) ) = ( F ` ( x S y ) ) )
28	6	subgoov 26114	. . . . . . 7 |- ( ( S e. ( SubGrpOp ` G ) /\ ( x e. Z /\ y e. Z ) ) -> ( x S y ) = ( x G y ) )
29	4, 28	sylan 479	. . . . . 6 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( x S y ) = ( x G y ) )
30	29	fveq2d 5883	. . . . 5 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( F ` ( x S y ) ) = ( F ` ( x G y ) ) )
31	27, 30	eqtrd 2505	. . . 4 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( ( F |` Z ) ` ( x S y ) ) = ( F ` ( x G y ) ) )
32		fvres 5893	. . . . . 6 |- ( x e. Z -> ( ( F |` Z ) ` x ) = ( F ` x ) )
33		fvres 5893	. . . . . 6 |- ( y e. Z -> ( ( F |` Z ) ` y ) = ( F ` y ) )
34	32, 33	oveqan12d 6327	. . . . 5 |- ( ( x e. Z /\ y e. Z ) -> ( ( ( F |` Z ) ` x ) O ( ( F |` Z ) ` y ) ) = ( ( F ` x ) O ( F ` y ) ) )
35	34	adantl 473	. . . 4 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( ( ( F |` Z ) ` x ) O ( ( F |` Z ) ` y ) ) = ( ( F ` x ) O ( F ` y ) ) )
36	19, 31, 35	3eqtr4d 2515	. . 3 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( ( F |` Z ) ` ( x S y ) ) = ( ( ( F |` Z ) ` x ) O ( ( F |` Z ) ` y ) ) )
37		ghsubgoOLD.9	. . . 4 |- H = ( O |` ( W X. W ) )
38		ghsubgoOLD.8	. . . . . 6 |- W = ( F " Z )
39	38, 38	xpeq12i 4861	. . . . 5 |- ( W X. W ) = ( ( F " Z ) X. ( F " Z ) )
40	39	reseq2i 5108	. . . 4 |- ( O |` ( W X. W ) ) = ( O |` ( ( F " Z ) X. ( F " Z ) ) )
41	37, 40	eqtri 2493	. . 3 |- H = ( O |` ( ( F " Z ) X. ( F " Z ) ) )
42		imassrn 5185	. . . . 5 |- ( F " Z ) C_ ran F
43		frn 5747	. . . . . 6 |- ( F : X --> Y -> ran F C_ Y )
44	1, 43	syl 17	. . . . 5 |- ( ph -> ran F C_ Y )
45	42, 44	syl5ss 3429	. . . 4 |- ( ph -> ( F " Z ) C_ Y )
46		ghsubgoOLD.4	. . . 4 |- ( ph -> Y C_ A )
47	45, 46	sstrd 3428	. . 3 |- ( ph -> ( F " Z ) C_ A )
48		ghsubgoOLD.5	. . 3 |- ( ph -> O Fn ( A X. A ) )
49	13, 36, 41, 6, 47, 48, 22	ghgrpOLD 26177	. 2 |- ( ph -> H e. GrpOp )
50	13	adantr 472	. . . 4 |- ( ( ph /\ S e. AbelOp ) -> ( F |` Z ) : Z -onto-> ( F " Z ) )
51	36	adantlr 729	. . . 4 |- ( ( ( ph /\ S e. AbelOp ) /\ ( x e. Z /\ y e. Z ) ) -> ( ( F |` Z ) ` ( x S y ) ) = ( ( ( F |` Z ) ` x ) O ( ( F |` Z ) ` y ) ) )
52	47	adantr 472	. . . 4 |- ( ( ph /\ S e. AbelOp ) -> ( F " Z ) C_ A )
53	48	adantr 472	. . . 4 |- ( ( ph /\ S e. AbelOp ) -> O Fn ( A X. A ) )
54		simpr 468	. . . 4 |- ( ( ph /\ S e. AbelOp ) -> S e. AbelOp )
55	50, 51, 41, 6, 52, 53, 54	ghabloOLD 26178	. . 3 |- ( ( ph /\ S e. AbelOp ) -> H e. AbelOp )
56	55	ex 441	. 2 |- ( ph -> ( S e. AbelOp -> H e. AbelOp ) )
57	49, 56	jca 541	1 |- ( ph -> ( H e. GrpOp /\ ( S e. AbelOp -> H e. AbelOp ) ) )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   C_ wss 3390   X. cxp 4837   dom cdm 4839   ran crn 4840   |` cres 4841   "cima 4842   Fun wfun 5583   Fn wfn 5584   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   GrpOpcgr 25995   AbelOpcablo 26090   SubGrpOpcsubgo 26110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-grpo 26000  df-gid 26001  df-ginv 26002  df-gdiv 26003  df-ablo 26091  df-subgo 26111

This theorem is referenced by:  ghsubgoOLD  26180  ghsubabloOLD  26181