Theorem resghm 13910

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 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (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 13827	. . 3 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 ∈ Grp)
7	6	adantl 277	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 ∈ Grp)
8		ghmgrp2 13896	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
9	8	adantr 276	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑇 ∈ Grp)
10		eqid 2231	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
11	10, 2	ghmf 13897	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	10	subgss 13824	. . . 4 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
13		fssres 5520	. . . 4 ⊢ ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
14	11, 12, 13	syl2an 289	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
15	5	a1i 9	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 = (𝑆 ↾s 𝑋))
16		eqidd 2232	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (Base‘𝑆) = (Base‘𝑆))
17		subgrcl 13829	. . . . . 6 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑆 ∈ Grp)
18	17	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑆 ∈ Grp)
19	12	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
20	15, 16, 18, 19	ressbas2d 13214	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 = (Base‘𝑈))
21	20	feq2d 5477	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
22	14, 21	mpbid 147	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇))
23		eleq2 2295	. . . . . 6 ⊢ (𝑋 = (Base‘𝑈) → (𝑎 ∈ 𝑋 ↔ 𝑎 ∈ (Base‘𝑈)))
24		eleq2 2295	. . . . . 6 ⊢ (𝑋 = (Base‘𝑈) → (𝑏 ∈ 𝑋 ↔ 𝑏 ∈ (Base‘𝑈)))
25	23, 24	anbi12d 473	. . . . 5 ⊢ (𝑋 = (Base‘𝑈) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))))
26	20, 25	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))))
27	26	biimpar 297	. . 3 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋))
28		simpll 527	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
29	19	sselda 3228	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (Base‘𝑆))
30	29	adantrr 479	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ (Base‘𝑆))
31	19	sselda 3228	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑏 ∈ 𝑋) → 𝑏 ∈ (Base‘𝑆))
32	31	adantrl 478	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ (Base‘𝑆))
33		eqid 2231	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
34	10, 33, 4	ghmlin 13898	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g‘𝑆)𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
35	28, 30, 32, 34	syl3anc 1274	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐹‘(𝑎(+g‘𝑆)𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
36	5	a1i 9	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 = (𝑆 ↾s 𝑋))
37		eqidd 2232	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → (+g‘𝑆) = (+g‘𝑆))
38		id 19	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ∈ (SubGrp‘𝑆))
39	36, 37, 38, 17	ressplusgd 13275	. . . . . . . 8 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → (+g‘𝑆) = (+g‘𝑈))
40	39	ad2antlr 489	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (+g‘𝑆) = (+g‘𝑈))
41	40	oveqd 6045	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) = (𝑎(+g‘𝑈)𝑏))
42	41	fveq2d 5652	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑎(+g‘𝑆)𝑏)) = ((𝐹 ↾ 𝑋)‘(𝑎(+g‘𝑈)𝑏)))
43	33	subgcl 13834	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubGrp‘𝑆) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
44	43	3expb 1231	. . . . . . 7 ⊢ ((𝑋 ∈ (SubGrp‘𝑆) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
45	44	adantll 476	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
46	45	fvresd 5673	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑎(+g‘𝑆)𝑏)) = (𝐹‘(𝑎(+g‘𝑆)𝑏)))
47	42, 46	eqtr3d 2266	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑎(+g‘𝑈)𝑏)) = (𝐹‘(𝑎(+g‘𝑆)𝑏)))
48		fvres 5672	. . . . . 6 ⊢ (𝑎 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑎) = (𝐹‘𝑎))
49		fvres 5672	. . . . . 6 ⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏))
50	48, 49	oveqan12d 6047	. . . . 5 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (((𝐹 ↾ 𝑋)‘𝑎)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
51	50	adantl 277	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐹 ↾ 𝑋)‘𝑎)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
52	35, 47, 51	3eqtr4d 2274	. . 3 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑎(+g‘𝑈)𝑏)) = (((𝐹 ↾ 𝑋)‘𝑎)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)))
53	27, 52	syldan 282	. 2 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (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 13902	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202   ⊆ wss 3201   ↾ cres 4733  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  Basecbs 13145   ↾_s cress 13146  +_gcplusg 13223  Grpcgrp 13646  SubGrpcsubg 13817   GrpHom cghm 13890

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-subg 13820  df-ghm 13891

This theorem is referenced by:  ghmima  13915  resrhm  14326