Theorem resghm 13846

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 2231	. 2 |- ( Base ` U ) = ( Base ` U )
2		eqid 2231	. 2 |- ( Base ` T ) = ( Base ` T )
3		eqid 2231	. 2 |- ( +g ` U ) = ( +g ` U )
4		eqid 2231	. 2 |- ( +g ` T ) = ( +g ` T )
5		resghm.u	. . . 4 |- U = ( S ↾s X )
6	5	subggrp 13763	. . 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 13832	. . 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 2231	. . . . 5 |- ( Base ` S ) = ( Base ` S )
11	10, 2	ghmf 13833	. . . 4 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
12	10	subgss 13760	. . . 4 |- ( X e. (SubGrp ` S ) -> X C_ ( Base ` S ) )
13		fssres 5512	. . . 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 2232	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( Base ` S ) = ( Base ` S ) )
17		subgrcl 13765	. . . . . 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 13150	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> X = ( Base ` U ) )
21	20	feq2d 5470	. . 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 2295	. . . . . 6 |- ( X = ( Base ` U ) -> ( a e. X <-> a e. ( Base ` U ) ) )
24		eleq2 2295	. . . . . 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 3227	. . . . . 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 3227	. . . . . 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 2231	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
34	10, 33, 4	ghmlin 13834	. . . . 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 1273	. . . 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 2232	. . . . . . . . 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 13211	. . . . . . . 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 6034	. . . . . 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 5643	. . . . 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 13770	. . . . . . . 8 |- ( ( X e. (SubGrp ` S ) /\ a e. X /\ b e. X ) -> ( a ( +g ` S ) b ) e. X )
44	43	3expb 1230	. . . . . . 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 5664	. . . . 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 2266	. . . 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 5663	. . . . . 6 |- ( a e. X -> ( ( F |` X ) ` a ) = ( F ` a ) )
49		fvres 5663	. . . . . 6 |- ( b e. X -> ( ( F |` X ) ` b ) = ( F ` b ) )
50	48, 49	oveqan12d 6036	. . . . 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 2274	. . 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 13838	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 1397   e. wcel 2202   C_ wss 3200   |` cres 4727   -->wf 5322   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   +g cplusg 13159   Grpcgrp 13582  SubGrpcsubg 13753   GrpHom cghm 13826

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-subg 13756  df-ghm 13827

This theorem is referenced by:  ghmima  13851  resrhm  14261