Theorem reslmhm 21047

Description: Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Hypotheses

Ref	Expression
reslmhm.u	⊢ 𝑈 = (LSubSp‘𝑆)
reslmhm.r	⊢ 𝑅 = (𝑆 ↾s 𝑋)

Assertion

Ref	Expression
reslmhm	⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 LMHom 𝑇))

Proof of Theorem reslmhm

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmlmod1 21028	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2		reslmhm.r	. . . 4 ⊢ 𝑅 = (𝑆 ↾s 𝑋)
3		reslmhm.u	. . . 4 ⊢ 𝑈 = (LSubSp‘𝑆)
4	2, 3	lsslmod 20955	. . 3 ⊢ ((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod)
5	1, 4	sylan 581	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod)
6		lmhmlmod2 21027	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7	6	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑇 ∈ LMod)
8		lmghm 21026	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9	3	lsssubg 20952	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
10	1, 9	sylan 581	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
11	2	resghm 19207	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇))
12	8, 10, 11	syl2an2r 686	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇))
13		eqid 2736	. . . . 5 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
14		eqid 2736	. . . . 5 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
15	13, 14	lmhmsca 21025	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
16	2, 13	resssca 17306	. . . 4 ⊢ (𝑋 ∈ 𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅))
17	15, 16	sylan9eq 2791	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Scalar‘𝑇) = (Scalar‘𝑅))
18		simpll 767	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
19		simprl 771	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
20		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	20, 3	lssss 20931	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ⊆ (Base‘𝑆))
22	21	adantl 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 ⊆ (Base‘𝑆))
23	22	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ⊆ (Base‘𝑆))
24	2, 20	ressbas2 17208	. . . . . . . . . . . 12 ⊢ (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑅))
25	22, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 = (Base‘𝑅))
26	25	eleq2d 2822	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝑏 ∈ 𝑋 ↔ 𝑏 ∈ (Base‘𝑅)))
27	26	biimpar 477	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ 𝑋)
28	27	adantrl 717	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ 𝑋)
29	23, 28	sseldd 3922	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆))
30		eqid 2736	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31		eqid 2736	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
32		eqid 2736	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
33	13, 30, 20, 31, 32	lmhmlin 21030	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
34	18, 19, 29, 33	syl3anc 1374	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
35	1	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑆 ∈ LMod)
36	35	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ LMod)
37		simplr 769	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ∈ 𝑈)
38	13, 31, 30, 3	lssvscl 20950	. . . . . . . 8 ⊢ (((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ 𝑋)) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ 𝑋)
39	36, 37, 19, 28, 38	syl22anc 839	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ 𝑋)
40	39	fvresd 6860	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)))
41		fvres 6859	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏))
42	41	oveq2d 7383	. . . . . . 7 ⊢ (𝑏 ∈ 𝑋 → (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
43	28, 42	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
44	34, 40, 43	3eqtr4d 2781	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)))
45	44	ralrimivva 3180	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)))
46	16	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Scalar‘𝑆) = (Scalar‘𝑅))
47	46	fveq2d 6844	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑅)))
48	2, 31	ressvsca 17307	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑈 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑅))
49	48	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑅))
50	49	oveqd 7384	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝑎( ·𝑠 ‘𝑆)𝑏) = (𝑎( ·𝑠 ‘𝑅)𝑏))
51	50	fveqeq2d 6848	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) ↔ ((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑅)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏))))
52	51	ralbidv 3160	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) ↔ ∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑅)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏))))
53	47, 52	raleqbidv 3311	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) ↔ ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑅)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏))))
54	45, 53	mpbid 232	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑅)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)))
55	12, 17, 54	3jca 1129	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → ((𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑅) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑅)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏))))
56		eqid 2736	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
57		eqid 2736	. . 3 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
58		eqid 2736	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
59		eqid 2736	. . 3 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
60	56, 14, 57, 58, 59, 32	islmhm 21022	. 2 ⊢ ((𝐹 ↾ 𝑋) ∈ (𝑅 LMHom 𝑇) ↔ ((𝑅 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ ((𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑅) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠 ‘𝑅)𝑏)) = (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)))))
61	5, 7, 55, 60	syl21anbrc 1346	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 LMHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889   ↾ cres 5633  ‘cfv 6498  (class class class)co 7367  Basecbs 17179   ↾_s cress 17200  Scalarcsca 17223   ·_𝑠 cvsca 17224  SubGrpcsubg 19096   GrpHom cghm 19187  LModclmod 20855  LSubSpclss 20926   LMHom clmhm 21014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-sca 17236  df-vsca 17237  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-ghm 19188  df-mgp 20122  df-ur 20163  df-ring 20216  df-lmod 20857  df-lss 20927  df-lmhm 21017

This theorem is referenced by:  frlmsplit2  21753  dimkerim  33771  lmhmlnmsplit  43515  pwssplit4  43517