Theorem resghm 19358

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 2762	. 2 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		eqid 2762	. 2 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2762	. 2 ⊢ (+g‘𝑈) = (+g‘𝑈)
4		eqid 2762	. 2 ⊢ (+g‘𝑇) = (+g‘𝑇)
5		resghm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
6	5	subggrp 19251	. . 3 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑈 ∈ Grp)
7	6	adantl 487	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑈 ∈ Grp)
8		ghmgrp2 19345	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
9	8	adantr 486	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑇 ∈ Grp)
10		eqid 2762	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
11	10, 2	ghmf 19346	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	10	subgss 19249	. . . 4 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
13		fssres 6745	. . . 4 ⊢ ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
14	11, 12, 13	syl2an 608	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
15	12	adantl 487	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
16	5, 10	ressbas2 17332	. . . . 5 ⊢ (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
17	15, 16	syl 18	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → 𝑋 = (Base‘𝑈))
18	17	feq2d 6690	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
19	14, 18	mpbid 235	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇))
20		eleq2 2851	. . . . . 6 ⊢ (𝑋 = (Base‘𝑈) → (𝑎 ∈ 𝑋 ↔ 𝑎 ∈ (Base‘𝑈)))
21		eleq2 2851	. . . . . 6 ⊢ (𝑋 = (Base‘𝑈) → (𝑏 ∈ 𝑋 ↔ 𝑏 ∈ (Base‘𝑈)))
22	20, 21	anbi12d 644	. . . . 5 ⊢ (𝑋 = (Base‘𝑈) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))))
23	17, 22	syl 18	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))))
24	23	biimpar 483	. . 3 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ (Base‘𝑈) ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋))
25		simpll 779	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
26	15	sselda 3934	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (Base‘𝑆))
27	26	adantrr 730	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ (Base‘𝑆))
28	15	sselda 3934	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ 𝑏 ∈ 𝑋) → 𝑏 ∈ (Base‘𝑆))
29	28	adantrl 729	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ (Base‘𝑆))
30		eqid 2762	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
31	10, 30, 4	ghmlin 19347	. . . . 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 1398	. . . 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 17378	. . . . . . . 8 ⊢ (𝑋 ∈ (SubGrp‘𝑆) → (+g‘𝑆) = (+g‘𝑈))
34	33	ad2antlr 740	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (+g‘𝑆) = (+g‘𝑈))
35	34	oveqd 7433	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) = (𝑎(+g‘𝑈)𝑏))
36	35	fveq2d 6886	. . . . 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 19258	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubGrp‘𝑆) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
38	37	3expb 1138	. . . . . . 7 ⊢ ((𝑋 ∈ (SubGrp‘𝑆) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
39	38	adantll 727	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(+g‘𝑆)𝑏) ∈ 𝑋)
40	39	fvresd 6902	. . . . 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 2799	. . . 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 6901	. . . . . 6 ⊢ (𝑎 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑎) = (𝐹‘𝑎))
43		fvres 6901	. . . . . 6 ⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏))
44	42, 43	oveqan12d 7435	. . . . 5 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (((𝐹 ↾ 𝑋)‘𝑎)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
45	44	adantl 487	. . . 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 2807	. . 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 603	. 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 19351	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3902   ↾ cres 5661  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  Basecbs 17303   ↾_s cress 17324  +_gcplusg 17344  Grpcgrp 19056  SubGrpcsubg 19242   GrpHom cghm 19339

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-sets 17258  df-slot 17276  df-ndx 17288  df-base 17304  df-ress 17325  df-plusg 17357  df-mgm 18732  df-sgrp 18821  df-mnd 18837  df-grp 19059  df-subg 19245  df-ghm 19340

This theorem is used by:  ghmima  19363  resrhm  20762  reslmhm  21235  dimkerim  34122