Theorem resghm 14040

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 2238	. 2 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		eqid 2238	. 2 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2238	. 2 ⊢ (+g‘𝑈) = (+g‘𝑈)
4		eqid 2238	. 2 ⊢ (+g‘𝑇) = (+g‘𝑇)
5		resghm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
6	5	subggrp 13957	. . 3 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 ∈ Grp)
7	6	adantl 277	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 ∈ Grp)
8		ghmgrp2 14026	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
9	8	adantr 276	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑇 ∈ Grp)
10		eqid 2238	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
11	10, 2	ghmf 14027	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	10	subgss 13954	. . . 4 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
13		fssres 5560	. . . 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 2239	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (Base‘𝑆) = (Base‘𝑆))
17		subgrcl 13959	. . . . . 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 13399	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 = (Base‘𝑈))
21	20	feq2d 5516	. . 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 2302	. . . . . 6 ⊢ (𝑋 = (Base‘𝑈) → (𝑎 ∈ 𝑋 ↔ 𝑎 ∈ (Base‘𝑈)))
24		eleq2 2302	. . . . . 6 ⊢ (𝑋 = (Base‘𝑈) → (𝑏 ∈ 𝑋 ↔ 𝑏 ∈ (Base‘𝑈)))
25	23, 24	anbi12d 477	. . . . 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 531	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
29	19	sselda 3248	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (Base‘𝑆))
30	29	adantrr 483	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ (Base‘𝑆))
31	19	sselda 3248	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑏 ∈ 𝑋) → 𝑏 ∈ (Base‘𝑆))
32	31	adantrl 482	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ (Base‘𝑆))
33		eqid 2238	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
34	10, 33, 4	ghmlin 14028	. . . . 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 1278	. . . 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 2239	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → (+g‘𝑆) = (+g‘𝑆))
38		id 19	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ∈ (SubGrp‘𝑆))
39	36, 37, 38, 17	ressplusgd 13460	. . . . . . . 8 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → (+g‘𝑆) = (+g‘𝑈))
40	39	ad2antlr 493	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (+g‘𝑆) = (+g‘𝑈))
41	40	oveqd 6092	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) = (𝑎(+g‘𝑈)𝑏))
42	41	fveq2d 5694	. . . . 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 13964	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubGrp‘𝑆) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
44	43	3expb 1235	. . . . . . 7 ⊢ ((𝑋 ∈ (SubGrp‘𝑆) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
45	44	adantll 480	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
46	45	fvresd 5715	. . . . 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 2273	. . . 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 5714	. . . . . 6 ⊢ (𝑎 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑎) = (𝐹‘𝑎))
49		fvres 5714	. . . . . 6 ⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏))
50	48, 49	oveqan12d 6094	. . . . 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 2281	. . 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 14032	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220   ↾ cres 4771  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  Basecbs 13330   ↾_s cress 13331  +_gcplusg 13408  Grpcgrp 13782  SubGrpcsubg 13947   GrpHom cghm 14020

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-subg 13950  df-ghm 14021

This theorem is referenced by:  ghmima  14045  resrhm  14529