Theorem ghsubgolemOLD 26098

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 5731	. . . . 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 26034	. . . . . 6 |- ( S e. ( SubGrpOp ` G ) -> Z C_ X )
8	4, 7	syl 17	. . . . 5 |- ( ph -> Z C_ X )
9		fdm 5733	. . . . . 6 |- ( F : X --> Y -> dom F = X )
10	1, 9	syl 17	. . . . 5 |- ( ph -> dom F = X )
11	8, 10	sseqtr4d 3469	. . . 4 |- ( ph -> Z C_ dom F )
12		fores 5802	. . . 4 |- ( ( Fun F /\ Z C_ dom F ) -> ( F |` Z ) : Z -onto-> ( F " Z ) )
13	3, 11, 12	syl2anc 667	. . 3 |- ( ph -> ( F |` Z ) : Z -onto-> ( F " Z ) )
14		ssel2 3427	. . . . . . 7 |- ( ( Z C_ X /\ x e. Z ) -> x e. X )
15		ssel2 3427	. . . . . . 7 |- ( ( Z C_ X /\ y e. Z ) -> y e. X )
16	14, 15	anim12dan 848	. . . . . 6 |- ( ( Z C_ X /\ ( x e. Z /\ y e. Z ) ) -> ( x e. X /\ y e. X ) )
17	8, 16	sylan 474	. . . . 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 473	. . . 4 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( F ` ( x G y ) ) = ( ( F ` x ) O ( F ` y ) ) )
20		issubgo 26031	. . . . . . . . 9 |- ( S e. ( SubGrpOp ` G ) <-> ( G e. GrpOp /\ S e. GrpOp /\ S C_ G ) )
21	20	simp2bi 1024	. . . . . . . 8 |- ( S e. ( SubGrpOp ` G ) -> S e. GrpOp )
22	4, 21	syl 17	. . . . . . 7 |- ( ph -> S e. GrpOp )
23	6	grpocl 25928	. . . . . . . 8 |- ( ( S e. GrpOp /\ x e. Z /\ y e. Z ) -> ( x S y ) e. Z )
24	23	3expb 1209	. . . . . . 7 |- ( ( S e. GrpOp /\ ( x e. Z /\ y e. Z ) ) -> ( x S y ) e. Z )
25	22, 24	sylan 474	. . . . . 6 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( x S y ) e. Z )
26		fvres 5879	. . . . . 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 26033	. . . . . . 7 |- ( ( S e. ( SubGrpOp ` G ) /\ ( x e. Z /\ y e. Z ) ) -> ( x S y ) = ( x G y ) )
29	4, 28	sylan 474	. . . . . 6 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( x S y ) = ( x G y ) )
30	29	fveq2d 5869	. . . . 5 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( F ` ( x S y ) ) = ( F ` ( x G y ) ) )
31	27, 30	eqtrd 2485	. . . 4 |- ( ( ph /\ ( x e. Z /\ y e. Z ) ) -> ( ( F |` Z ) ` ( x S y ) ) = ( F ` ( x G y ) ) )
32		fvres 5879	. . . . . 6 |- ( x e. Z -> ( ( F |` Z ) ` x ) = ( F ` x ) )
33		fvres 5879	. . . . . 6 |- ( y e. Z -> ( ( F |` Z ) ` y ) = ( F ` y ) )
34	32, 33	oveqan12d 6309	. . . . 5 |- ( ( x e. Z /\ y e. Z ) -> ( ( ( F |` Z ) ` x ) O ( ( F |` Z ) ` y ) ) = ( ( F ` x ) O ( F ` y ) ) )
35	34	adantl 468	. . . 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 2495	. . 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 4856	. . . . 5 |- ( W X. W ) = ( ( F " Z ) X. ( F " Z ) )
40	39	reseq2i 5102	. . . 4 |- ( O |` ( W X. W ) ) = ( O |` ( ( F " Z ) X. ( F " Z ) ) )
41	37, 40	eqtri 2473	. . 3 |- H = ( O |` ( ( F " Z ) X. ( F " Z ) ) )
42		imassrn 5179	. . . . 5 |- ( F " Z ) C_ ran F
43		frn 5735	. . . . . 6 |- ( F : X --> Y -> ran F C_ Y )
44	1, 43	syl 17	. . . . 5 |- ( ph -> ran F C_ Y )
45	42, 44	syl5ss 3443	. . . 4 |- ( ph -> ( F " Z ) C_ Y )
46		ghsubgoOLD.4	. . . 4 |- ( ph -> Y C_ A )
47	45, 46	sstrd 3442	. . 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 26096	. 2 |- ( ph -> H e. GrpOp )
50	13	adantr 467	. . . 4 |- ( ( ph /\ S e. AbelOp ) -> ( F |` Z ) : Z -onto-> ( F " Z ) )
51	36	adantlr 721	. . . 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 467	. . . 4 |- ( ( ph /\ S e. AbelOp ) -> ( F " Z ) C_ A )
53	48	adantr 467	. . . 4 |- ( ( ph /\ S e. AbelOp ) -> O Fn ( A X. A ) )
54		simpr 463	. . . 4 |- ( ( ph /\ S e. AbelOp ) -> S e. AbelOp )
55	50, 51, 41, 6, 52, 53, 54	ghabloOLD 26097	. . 3 |- ( ( ph /\ S e. AbelOp ) -> H e. AbelOp )
56	55	ex 436	. 2 |- ( ph -> ( S e. AbelOp -> H e. AbelOp ) )
57	49, 56	jca 535	1 |- ( ph -> ( H e. GrpOp /\ ( S e. AbelOp -> H e. AbelOp ) ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   C_ wss 3404   X. cxp 4832   dom cdm 4834   ran crn 4835   |` cres 4836   "cima 4837   Fun wfun 5576   Fn wfn 5577   -->wf 5578   -onto->wfo 5580   ` cfv 5582  (class class class)co 6290   GrpOpcgr 25914   AbelOpcablo 26009   SubGrpOpcsubgo 26029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-grpo 25919  df-gid 25920  df-ginv 25921  df-gdiv 25922  df-ablo 26010  df-subgo 26030

This theorem is referenced by:  ghsubgoOLD  26099  ghsubabloOLD  26100