Theorem resghm 13466

Description: Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Hypothesis

Ref	Expression
resghm.u	|- U = ( S ↾s X )

Assertion

Ref	Expression
resghm	|- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( F |` X ) e. ( U GrpHom T ) )

Proof of Theorem resghm

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2196	. 2 |- ( Base ` U ) = ( Base ` U )
2		eqid 2196	. 2 |- ( Base ` T ) = ( Base ` T )
3		eqid 2196	. 2 |- ( +g ` U ) = ( +g ` U )
4		eqid 2196	. 2 |- ( +g ` T ) = ( +g ` T )
5		resghm.u	. . . 4 |- U = ( S ↾s X )
6	5	subggrp 13383	. . 3 |- ( X e. (SubGrp ` S ) -> U e. Grp )
7	6	adantl 277	. 2 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> U e. Grp )
8		ghmgrp2 13452	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Grp )
9	8	adantr 276	. 2 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> T e. Grp )
10		eqid 2196	. . . . 5 |- ( Base ` S ) = ( Base ` S )
11	10, 2	ghmf 13453	. . . 4 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
12	10	subgss 13380	. . . 4 |- ( X e. (SubGrp ` S ) -> X C_ ( Base ` S ) )
13		fssres 5436	. . . 4 |- ( ( F : ( Base ` S ) --> ( Base ` T ) /\ X C_ ( Base ` S ) ) -> ( F |` X ) : X --> ( Base ` T ) )
14	11, 12, 13	syl2an 289	. . 3 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( F |` X ) : X --> ( Base ` T ) )
15	5	a1i 9	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> U = ( S ↾s X ) )
16		eqidd 2197	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( Base ` S ) = ( Base ` S ) )
17		subgrcl 13385	. . . . . 6 |- ( X e. (SubGrp ` S ) -> S e. Grp )
18	17	adantl 277	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> S e. Grp )
19	12	adantl 277	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> X C_ ( Base ` S ) )
20	15, 16, 18, 19	ressbas2d 12771	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> X = ( Base ` U ) )
21	20	feq2d 5398	. . 3 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( ( F |` X ) : X --> ( Base ` T ) <-> ( F |` X ) : ( Base ` U ) --> ( Base ` T ) ) )
22	14, 21	mpbid 147	. 2 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( F |` X ) : ( Base ` U ) --> ( Base ` T ) )
23		eleq2 2260	. . . . . 6 |- ( X = ( Base ` U ) -> ( a e. X <-> a e. ( Base ` U ) ) )
24		eleq2 2260	. . . . . 6 |- ( X = ( Base ` U ) -> ( b e. X <-> b e. ( Base ` U ) ) )
25	23, 24	anbi12d 473	. . . . 5 |- ( X = ( Base ` U ) -> ( ( a e. X /\ b e. X ) <-> ( a e. ( Base ` U ) /\ b e. ( Base ` U ) ) ) )
26	20, 25	syl 14	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( ( a e. X /\ b e. X ) <-> ( a e. ( Base ` U ) /\ b e. ( Base ` U ) ) ) )
27	26	biimpar 297	. . 3 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. ( Base ` U ) /\ b e. ( Base ` U ) ) ) -> ( a e. X /\ b e. X ) )
28		simpll 527	. . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> F e. ( S GrpHom T ) )
29	19	sselda 3184	. . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ a e. X ) -> a e. ( Base ` S ) )
30	29	adantrr 479	. . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> a e. ( Base ` S ) )
31	19	sselda 3184	. . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ b e. X ) -> b e. ( Base ` S ) )
32	31	adantrl 478	. . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> b e. ( Base ` S ) )
33		eqid 2196	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
34	10, 33, 4	ghmlin 13454	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ a e. ( Base ` S ) /\ b e. ( Base ` S ) ) -> ( F ` ( a ( +g ` S ) b ) ) = ( ( F ` a ) ( +g ` T ) ( F ` b ) ) )
35	28, 30, 32, 34	syl3anc 1249	. . . 4 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> ( F ` ( a ( +g ` S ) b ) ) = ( ( F ` a ) ( +g ` T ) ( F ` b ) ) )
36	5	a1i 9	. . . . . . . . 9 |- ( X e. (SubGrp ` S ) -> U = ( S ↾s X ) )
37		eqidd 2197	. . . . . . . . 9 |- ( X e. (SubGrp ` S ) -> ( +g ` S ) = ( +g ` S ) )
38		id 19	. . . . . . . . 9 |- ( X e. (SubGrp ` S ) -> X e. (SubGrp ` S ) )
39	36, 37, 38, 17	ressplusgd 12831	. . . . . . . 8 |- ( X e. (SubGrp ` S ) -> ( +g ` S ) = ( +g ` U ) )
40	39	ad2antlr 489	. . . . . . 7 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> ( +g ` S ) = ( +g ` U ) )
41	40	oveqd 5942	. . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> ( a ( +g ` S ) b ) = ( a ( +g ` U ) b ) )
42	41	fveq2d 5565	. . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> ( ( F |` X ) ` ( a ( +g ` S ) b ) ) = ( ( F |` X ) ` ( a ( +g ` U ) b ) ) )
43	33	subgcl 13390	. . . . . . . 8 |- ( ( X e. (SubGrp ` S ) /\ a e. X /\ b e. X ) -> ( a ( +g ` S ) b ) e. X )
44	43	3expb 1206	. . . . . . 7 |- ( ( X e. (SubGrp ` S ) /\ ( a e. X /\ b e. X ) ) -> ( a ( +g ` S ) b ) e. X )
45	44	adantll 476	. . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> ( a ( +g ` S ) b ) e. X )
46	45	fvresd 5586	. . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> ( ( F |` X ) ` ( a ( +g ` S ) b ) ) = ( F ` ( a ( +g ` S ) b ) ) )
47	42, 46	eqtr3d 2231	. . . 4 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> ( ( F |` X ) ` ( a ( +g ` U ) b ) ) = ( F ` ( a ( +g ` S ) b ) ) )
48		fvres 5585	. . . . . 6 |- ( a e. X -> ( ( F |` X ) ` a ) = ( F ` a ) )
49		fvres 5585	. . . . . 6 |- ( b e. X -> ( ( F |` X ) ` b ) = ( F ` b ) )
50	48, 49	oveqan12d 5944	. . . . 5 |- ( ( a e. X /\ b e. X ) -> ( ( ( F |` X ) ` a ) ( +g ` T ) ( ( F |` X ) ` b ) ) = ( ( F ` a ) ( +g ` T ) ( F ` b ) ) )
51	50	adantl 277	. . . 4 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> ( ( ( F |` X ) ` a ) ( +g ` T ) ( ( F |` X ) ` b ) ) = ( ( F ` a ) ( +g ` T ) ( F ` b ) ) )
52	35, 47, 51	3eqtr4d 2239	. . 3 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. X /\ b e. X ) ) -> ( ( F |` X ) ` ( a ( +g ` U ) b ) ) = ( ( ( F |` X ) ` a ) ( +g ` T ) ( ( F |` X ) ` b ) ) )
53	27, 52	syldan 282	. 2 |- ( ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) /\ ( a e. ( Base ` U ) /\ b e. ( Base ` U ) ) ) -> ( ( F |` X ) ` ( a ( +g ` U ) b ) ) = ( ( ( F |` X ) ` a ) ( +g ` T ) ( ( F |` X ) ` b ) ) )
54	1, 2, 3, 4, 7, 9, 22, 53	isghmd 13458	1 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( F |` X ) e. ( U GrpHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   C_ wss 3157   |` cres 4666   -->wf 5255   ` cfv 5259  (class class class)co 5925   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   Grpcgrp 13202  SubGrpcsubg 13373   GrpHom cghm 13446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-subg 13376  df-ghm 13447

This theorem is referenced by:  ghmima  13471  resrhm  13880