Theorem reslmhm 21360

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 21341	. . 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 21268	. . 3 ⊢ ((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod)
5	1, 4	sylan 592	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod)
6		lmhmlmod2 21340	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7	6	adantr 486	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑇 ∈ LMod)
8		lmghm 21339	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9	3	lsssubg 21265	. . . . 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 19478	. . . 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 2763	. . . . 5 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
14		eqid 2763	. . . . 5 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
15	13, 14	lmhmsca 21338	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
16	2, 13	resssca 17546	. . . 4 ⊢ (𝑋 ∈ 𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅))
17	15, 16	sylan9eq 2818	. . 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 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	20, 3	lssss 21244	. . . . . . . . . 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 17448	. . . . . . . . . . . 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 2850	. . . . . . . . . 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 3914	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆))
30		eqid 2763	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31		eqid 2763	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
32		eqid 2763	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
33	13, 30, 20, 31, 32	lmhmlin 21343	. . . . . . 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 1399	. . . . . 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 21263	. . . . . . . 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 6931	. . . . . 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 6930	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏))
42	41	oveq2d 7462	. . . . . . 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 2808	. . . . 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 3209	. . . 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 6915	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑅)))
48	2, 31	ressvsca 17547	. . . . . . . . 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 7463	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝑎( ·𝑠 ‘𝑆)𝑏) = (𝑎( ·𝑠 ‘𝑅)𝑏))
51	50	fveqeq2d 6919	. . . . . 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 3189	. . . . 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 3338	. . . 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 2763	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
57		eqid 2763	. . 3 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
58		eqid 2763	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
59		eqid 2763	. . 3 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
60	56, 14, 57, 58, 59, 32	islmhm 21335	. 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 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ↾ cres 5658  ‘cfv 6561  (class class class)co 7446  Basecbs 17419   ↾_s cress 17440  Scalarcsca 17463   ·_𝑠 cvsca 17464  SubGrpcsubg 19362   GrpHom cghm 19459  LModclmod 21168  LSubSpclss 21239   LMHom clmhm 21327

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-3 12436  df-4 12437  df-5 12438  df-6 12439  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-ress 17441  df-plusg 17473  df-sca 17476  df-vsca 17477  df-0g 17644  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-grp 19179  df-minusg 19180  df-sbg 19181  df-subg 19365  df-ghm 19460  df-mgp 20394  df-ur 20441  df-ring 20494  df-lmod 21170  df-lss 21240  df-lmhm 21330

This theorem is used by:  frlmsplit2  22112  dimkerim  34342  lmhmlnmsplit  44293  pwssplit4  44295