Theorem resghm 13333

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 2193	. 2 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		eqid 2193	. 2 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2193	. 2 ⊢ (+g‘𝑈) = (+g‘𝑈)
4		eqid 2193	. 2 ⊢ (+g‘𝑇) = (+g‘𝑇)
5		resghm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
6	5	subggrp 13250	. . 3 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 ∈ Grp)
7	6	adantl 277	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 ∈ Grp)
8		ghmgrp2 13319	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
9	8	adantr 276	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑇 ∈ Grp)
10		eqid 2193	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
11	10, 2	ghmf 13320	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	10	subgss 13247	. . . 4 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
13		fssres 5430	. . . 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 2194	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (Base‘𝑆) = (Base‘𝑆))
17		subgrcl 13252	. . . . . 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 12689	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 = (Base‘𝑈))
21	20	feq2d 5392	. . 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 2257	. . . . . 6 ⊢ (𝑋 = (Base‘𝑈) → (𝑎 ∈ 𝑋 ↔ 𝑎 ∈ (Base‘𝑈)))
24		eleq2 2257	. . . . . 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 3180	. . . . . 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 3180	. . . . . 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 2193	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
34	10, 33, 4	ghmlin 13321	. . . . 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 1249	. . . 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 2194	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → (+g‘𝑆) = (+g‘𝑆))
38		id 19	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ∈ (SubGrp‘𝑆))
39	36, 37, 38, 17	ressplusgd 12749	. . . . . . . 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 5936	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) = (𝑎(+g‘𝑈)𝑏))
42	41	fveq2d 5559	. . . . 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 13257	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubGrp‘𝑆) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
44	43	3expb 1206	. . . . . . 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 5580	. . . . 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 2228	. . . 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 5579	. . . . . 6 ⊢ (𝑎 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑎) = (𝐹‘𝑎))
49		fvres 5579	. . . . . 6 ⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏))
50	48, 49	oveqan12d 5938	. . . . 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 2236	. . 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 13325	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164   ⊆ wss 3154   ↾ cres 4662  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919  Basecbs 12621   ↾_s cress 12622  +_gcplusg 12698  Grpcgrp 13075  SubGrpcsubg 13240   GrpHom cghm 13313

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-subg 13243  df-ghm 13314

This theorem is referenced by:  ghmima  13338  resrhm  13747