Theorem resghm 19202

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 2741	. 2 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		eqid 2741	. 2 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2741	. 2 ⊢ (+g‘𝑈) = (+g‘𝑈)
4		eqid 2741	. 2 ⊢ (+g‘𝑇) = (+g‘𝑇)
5		resghm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
6	5	subggrp 19100	. . 3 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 ∈ Grp)
7	6	adantl 483	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 ∈ Grp)
8		ghmgrp2 19189	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
9	8	adantr 482	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑇 ∈ Grp)
10		eqid 2741	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
11	10, 2	ghmf 19190	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	10	subgss 19098	. . . 4 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
13		fssres 6697	. . . 4 ⊢ ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
14	11, 12, 13	syl2an 603	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
15	12	adantl 483	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
16	5, 10	ressbas2 17203	. . . . 5 ⊢ (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
17	15, 16	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 = (Base‘𝑈))
18	17	feq2d 6643	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
19	14, 18	mpbid 234	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇))
20		eleq2 2830	. . . . . 6 ⊢ (𝑋 = (Base‘𝑈) → (𝑎 ∈ 𝑋 ↔ 𝑎 ∈ (Base‘𝑈)))
21		eleq2 2830	. . . . . 6 ⊢ (𝑋 = (Base‘𝑈) → (𝑏 ∈ 𝑋 ↔ 𝑏 ∈ (Base‘𝑈)))
22	20, 21	anbi12d 639	. . . . 5 ⊢ (𝑋 = (Base‘𝑈) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))))
23	17, 22	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))))
24	23	biimpar 479	. . 3 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋))
25		simpll 773	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
26	15	sselda 3917	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (Base‘𝑆))
27	26	adantrr 724	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ (Base‘𝑆))
28	15	sselda 3917	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑏 ∈ 𝑋) → 𝑏 ∈ (Base‘𝑆))
29	28	adantrl 723	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ (Base‘𝑆))
30		eqid 2741	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
31	10, 30, 4	ghmlin 19191	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g‘𝑆)𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
32	25, 27, 29, 31	syl3anc 1380	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐹‘(𝑎(+g‘𝑆)𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
33	5, 30	ressplusg 17249	. . . . . . . 8 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → (+g‘𝑆) = (+g‘𝑈))
34	33	ad2antlr 734	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (+g‘𝑆) = (+g‘𝑈))
35	34	oveqd 7377	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) = (𝑎(+g‘𝑈)𝑏))
36	35	fveq2d 6835	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑎(+g‘𝑆)𝑏)) = ((𝐹 ↾ 𝑋)‘(𝑎(+g‘𝑈)𝑏)))
37	30	subgcl 19107	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubGrp‘𝑆) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
38	37	3expb 1127	. . . . . . 7 ⊢ ((𝑋 ∈ (SubGrp‘𝑆) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
39	38	adantll 721	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
40	39	fvresd 6851	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑎(+g‘𝑆)𝑏)) = (𝐹‘(𝑎(+g‘𝑆)𝑏)))
41	36, 40	eqtr3d 2778	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑎(+g‘𝑈)𝑏)) = (𝐹‘(𝑎(+g‘𝑆)𝑏)))
42		fvres 6850	. . . . . 6 ⊢ (𝑎 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑎) = (𝐹‘𝑎))
43		fvres 6850	. . . . . 6 ⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏))
44	42, 43	oveqan12d 7379	. . . . 5 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (((𝐹 ↾ 𝑋)‘𝑎)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
45	44	adantl 483	. . . 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)))
46	32, 41, 45	3eqtr4d 2786	. . 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)))
47	24, 46	syldan 598	. 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)))
48	1, 2, 3, 4, 7, 9, 19, 47	isghmd 19195	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ⊆ wss 3885   ↾ cres 5623  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  Basecbs 17174   ↾_s cress 17195  +_gcplusg 17215  Grpcgrp 18904  SubGrpcsubg 19091   GrpHom cghm 19182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18907  df-subg 19094  df-ghm 19183

This theorem is referenced by:  ghmima  19207  resrhm  20577  reslmhm  21046  dimkerim  33823