Theorem reslmhm 20229

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 20210	. . 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 20137	. . 3 ⊢ ((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod)
5	1, 4	sylan 579	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod)
6		lmhmlmod2 20209	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7	6	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑇 ∈ LMod)
8		lmghm 20208	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9	3	lsssubg 20134	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
10	1, 9	sylan 579	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
11	2	resghm 18765	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇))
12	8, 10, 11	syl2an2r 681	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇))
13		eqid 2738	. . . . 5 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
14		eqid 2738	. . . . 5 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
15	13, 14	lmhmsca 20207	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
16	2, 13	resssca 16978	. . . 4 ⊢ (𝑋 ∈ 𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅))
17	15, 16	sylan9eq 2799	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Scalar‘𝑇) = (Scalar‘𝑅))
18		simpll 763	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
19		simprl 767	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
20		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	20, 3	lssss 20113	. . . . . . . . . 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 16875	. . . . . . . . . . . 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 2824	. . . . . . . . . 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 712	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ 𝑋)
29	23, 28	sseldd 3918	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆))
30		eqid 2738	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31		eqid 2738	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
32		eqid 2738	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
33	13, 30, 20, 31, 32	lmhmlin 20212	. . . . . . 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 1369	. . . . . 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 765	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ∈ 𝑈)
38	13, 31, 30, 3	lssvscl 20132	. . . . . . . 8 ⊢ (((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ 𝑋)) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ 𝑋)
39	36, 37, 19, 28, 38	syl22anc 835	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ 𝑋)
40	39	fvresd 6776	. . . . . 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 6775	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏))
42	41	oveq2d 7271	. . . . . . 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 2788	. . . . 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 3114	. . . 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 6760	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑅)))
48	2, 31	ressvsca 16979	. . . . . . . . 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 7272	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝑎( ·𝑠 ‘𝑆)𝑏) = (𝑎( ·𝑠 ‘𝑅)𝑏))
51	50	fveqeq2d 6764	. . . . . 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 3120	. . . . 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 3327	. . . 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 231	. . 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 1126	. 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 2738	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
57		eqid 2738	. . 3 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
58		eqid 2738	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
59		eqid 2738	. . 3 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
60	56, 14, 57, 58, 59, 32	islmhm 20204	. 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 1342	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 LMHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883   ↾ cres 5582  ‘cfv 6418  (class class class)co 7255  Basecbs 16840   ↾_s cress 16867  Scalarcsca 16891   ·_𝑠 cvsca 16892  SubGrpcsubg 18664   GrpHom cghm 18746  LModclmod 20038  LSubSpclss 20108   LMHom clmhm 20196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-sca 16904  df-vsca 16905  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-ghm 18747  df-mgp 19636  df-ur 19653  df-ring 19700  df-lmod 20040  df-lss 20109  df-lmhm 20199

This theorem is referenced by:  frlmsplit2  20890  dimkerim  31610  lmhmlnmsplit  40828  pwssplit4  40830