Theorem resghm 14013

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 2234	. 2 |- ( Base ` U ) = ( Base ` U )
2		eqid 2234	. 2 |- ( Base ` T ) = ( Base ` T )
3		eqid 2234	. 2 |- ( +g ` U ) = ( +g ` U )
4		eqid 2234	. 2 |- ( +g ` T ) = ( +g ` T )
5		resghm.u	. . . 4 |- U = ( S ↾s X )
6	5	subggrp 13930	. . 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 13999	. . 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 2234	. . . . 5 |- ( Base ` S ) = ( Base ` S )
11	10, 2	ghmf 14000	. . . 4 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
12	10	subgss 13927	. . . 4 |- ( X e. (SubGrp ` S ) -> X C_ ( Base ` S ) )
13		fssres 5545	. . . 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 2235	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( Base ` S ) = ( Base ` S ) )
17		subgrcl 13932	. . . . . 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 13365	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> X = ( Base ` U ) )
21	20	feq2d 5501	. . 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 2298	. . . . . 6 |- ( X = ( Base ` U ) -> ( a e. X <-> a e. ( Base ` U ) ) )
24		eleq2 2298	. . . . . 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 3242	. . . . . 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 3242	. . . . . 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 2234	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
34	10, 33, 4	ghmlin 14001	. . . . 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 1274	. . . 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 2235	. . . . . . . . 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 13426	. . . . . . . 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 6075	. . . . . 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 5679	. . . . 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 13937	. . . . . . . 8 |- ( ( X e. (SubGrp ` S ) /\ a e. X /\ b e. X ) -> ( a ( +g ` S ) b ) e. X )
44	43	3expb 1231	. . . . . . 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 5700	. . . . 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 2269	. . . 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 5699	. . . . . 6 |- ( a e. X -> ( ( F |` X ) ` a ) = ( F ` a ) )
49		fvres 5699	. . . . . 6 |- ( b e. X -> ( ( F |` X ) ` b ) = ( F ` b ) )
50	48, 49	oveqan12d 6077	. . . . 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 2277	. . 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 14005	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 1398   e. wcel 2205   C_ wss 3214   |` cres 4756   -->wf 5353   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   +g cplusg 13374   Grpcgrp 13755  SubGrpcsubg 13920   GrpHom cghm 13993

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-subg 13923  df-ghm 13994

This theorem is referenced by:  ghmima  14018  resrhm  14494