Theorem reslmhm 21332

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 21313	. . 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 21240	. . 3 ⊢ ((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod)
5	1, 4	sylan 592	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod)
6		lmhmlmod2 21312	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7	6	adantr 486	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑇 ∈ LMod)
8		lmghm 21311	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9	3	lsssubg 21237	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
10	1, 9	sylan 592	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
11	2	resghm 19451	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇))
12	8, 10, 11	syl2an2r 698	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇))
13		eqid 2760	. . . . 5 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
14		eqid 2760	. . . . 5 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
15	13, 14	lmhmsca 21310	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
16	2, 13	resssca 17519	. . . 4 ⊢ (𝑋 ∈ 𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅))
17	15, 16	sylan9eq 2815	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Scalar‘𝑇) = (Scalar‘𝑅))
18		simpll 779	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
19		simprl 783	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
20		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	20, 3	lssss 21216	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ⊆ (Base‘𝑆))
22	21	adantl 487	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 ⊆ (Base‘𝑆))
23	22	adantr 486	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ⊆ (Base‘𝑆))
24	2, 20	ressbas2 17421	. . . . . . . . . . . 12 ⊢ (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑅))
25	22, 24	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 = (Base‘𝑅))
26	25	eleq2d 2846	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝑏 ∈ 𝑋 ↔ 𝑏 ∈ (Base‘𝑅)))
27	26	biimpar 483	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ 𝑋)
28	27	adantrl 729	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ 𝑋)
29	23, 28	sseldd 3909	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆))
30		eqid 2760	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31		eqid 2760	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
32		eqid 2760	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
33	13, 30, 20, 31, 32	lmhmlin 21315	. . . . . . 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 1398	. . . . . 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 486	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑆 ∈ LMod)
36	35	adantr 486	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ LMod)
37		simplr 781	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ∈ 𝑈)
38	13, 31, 30, 3	lssvscl 21235	. . . . . . . 8 ⊢ (((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ 𝑋)) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ 𝑋)
39	36, 37, 19, 28, 38	syl22anc 852	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ 𝑋)
40	39	fvresd 6905	. . . . . 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 6904	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏))
42	41	oveq2d 7436	. . . . . . 7 ⊢ (𝑏 ∈ 𝑋 → (𝑎( ·𝑠 ‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
43	28, 42	syl 18	. . . . . 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 2805	. . . . 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 3205	. . . 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 487	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Scalar‘𝑆) = (Scalar‘𝑅))
47	46	fveq2d 6889	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑅)))
48	2, 31	ressvsca 17520	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑈 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑅))
49	48	adantl 487	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑅))
50	49	oveqd 7437	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝑎( ·𝑠 ‘𝑆)𝑏) = (𝑎( ·𝑠 ‘𝑅)𝑏))
51	50	fveqeq2d 6893	. . . . . 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 3185	. . . . 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 3334	. . . 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 235	. . 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 1146	. 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 2760	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
57		eqid 2760	. . 3 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
58		eqid 2760	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
59		eqid 2760	. . 3 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
60	56, 14, 57, 58, 59, 32	islmhm 21307	. 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 1363	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 LMHom 𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ↾ cres 5652  ‘cfv 6538  (class class class)co 7420  Basecbs 17392   ↾_s cress 17413  Scalarcsca 17436   ·_𝑠 cvsca 17437  SubGrpcsubg 19335   GrpHom cghm 19432  LModclmod 21140  LSubSpclss 21211   LMHom clmhm 21299

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-er 8720  df-map 8852  df-en 8977  df-dom 8978  df-sdom 8979  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-sca 17449  df-vsca 17450  df-0g 17617  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-grp 19152  df-minusg 19153  df-sbg 19154  df-subg 19338  df-ghm 19433  df-mgp 20366  df-ur 20413  df-ring 20466  df-lmod 21142  df-lss 21212  df-lmhm 21302

This theorem is used by:  frlmsplit2  22084  dimkerim  34305  lmhmlnmsplit  44182  pwssplit4  44184