frlmssuvc1 - Metamath Proof Explorer

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Theorem frlmssuvc1 21747

Description: A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)

**Hypotheses**
Ref	Expression
frlmssuvc1.f	⊢ 𝐹 = (𝑅 freeLMod 𝐼)
frlmssuvc1.u	⊢ 𝑈 = (𝑅 unitVec 𝐼)
frlmssuvc1.b	⊢ 𝐵 = (Base‘𝐹)
frlmssuvc1.k	⊢ 𝐾 = (Base‘𝑅)
frlmssuvc1.t	⊢ · = ( ·_𝑠 ‘𝐹)
frlmssuvc1.z	⊢ 0 = (0_g‘𝑅)
frlmssuvc1.c	⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}
frlmssuvc1.r	⊢ (𝜑 → 𝑅 ∈ Ring)
frlmssuvc1.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
frlmssuvc1.j	⊢ (𝜑 → 𝐽 ⊆ 𝐼)
frlmssuvc1.l	⊢ (𝜑 → 𝐿 ∈ 𝐽)
frlmssuvc1.x	⊢ (𝜑 → 𝑋 ∈ 𝐾)

**Assertion**
Ref	Expression
frlmssuvc1	⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)

Distinct variable groups: 𝑥,𝐵 𝑥,𝐹 𝑥,𝐼 𝑥,𝐽 𝑥,𝐾 𝑥,𝐿 𝑥,𝑅 𝑥, 0 𝜑,𝑥 𝑥,𝑈 𝑥,𝑉 𝑥, · 𝑥,𝑋 Allowed substitution hint: 𝐶(𝑥)

**Proof of Theorem frlmssuvc1**
Step	Hyp	Ref	Expression
1		frlmssuvc1.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		frlmssuvc1.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
3		frlmssuvc1.f	. . . 4 ⊢ 𝐹 = (𝑅 freeLMod 𝐼)
4	3	frlmlmod 21702	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ LMod)
5	1, 2, 4	syl2anc 584	. 2 ⊢ (𝜑 → 𝐹 ∈ LMod)
6		frlmssuvc1.j	. . 3 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
7		eqid 2734	. . . 4 ⊢ (LSubSp‘𝐹) = (LSubSp‘𝐹)
8		frlmssuvc1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
9		frlmssuvc1.z	. . . 4 ⊢ 0 = (0_g‘𝑅)
10		frlmssuvc1.c	. . . 4 ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}
11	3, 7, 8, 9, 10	frlmsslss2 21728	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ (LSubSp‘𝐹))
12	1, 2, 6, 11	syl3anc 1373	. 2 ⊢ (𝜑 → 𝐶 ∈ (LSubSp‘𝐹))
13		frlmssuvc1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
14		frlmssuvc1.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
15	3	frlmsca 21706	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
16	1, 2, 15	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
17	16	fveq2d 6836	. . . 4 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
18	14, 17	eqtrid 2781	. . 3 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝐹)))
19	13, 18	eleqtrd 2836	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝐹)))
20		frlmssuvc1.u	. . . . . 6 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
21	20, 3, 8	uvcff 21744	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶𝐵)
22	1, 2, 21	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑈:𝐼⟶𝐵)
23		frlmssuvc1.l	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ 𝐽)
24	6, 23	sseldd 3932	. . . 4 ⊢ (𝜑 → 𝐿 ∈ 𝐼)
25	22, 24	ffvelcdmd 7028	. . 3 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐵)
26	3, 14, 8	frlmbasf 21713	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑈‘𝐿) ∈ 𝐵) → (𝑈‘𝐿):𝐼⟶𝐾)
27	2, 25, 26	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑈‘𝐿):𝐼⟶𝐾)
28	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑅 ∈ Ring)
29	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐼 ∈ 𝑉)
30	24	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ∈ 𝐼)
31		eldifi 4081	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼)
32	31	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼)
33		disjdif 4422	. . . . . 6 ⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅
34		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ (𝐼 ∖ 𝐽))
35		disjne 4405	. . . . . 6 ⊢ (((𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ ∧ 𝐿 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ≠ 𝑥)
36	33, 23, 34, 35	mp3an2ani 1470	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ≠ 𝑥)
37	20, 28, 29, 30, 32, 36, 9	uvcvv0 21743	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝐿)‘𝑥) = 0 )
38	27, 37	suppss 8134	. . 3 ⊢ (𝜑 → ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽)
39		oveq1 7363	. . . . 5 ⊢ (𝑥 = (𝑈‘𝐿) → (𝑥 supp 0 ) = ((𝑈‘𝐿) supp 0 ))
40	39	sseq1d 3963	. . . 4 ⊢ (𝑥 = (𝑈‘𝐿) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽))
41	40, 10	elrab2 3647	. . 3 ⊢ ((𝑈‘𝐿) ∈ 𝐶 ↔ ((𝑈‘𝐿) ∈ 𝐵 ∧ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽))
42	25, 38, 41	sylanbrc 583	. 2 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐶)
43		eqid 2734	. . 3 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
44		frlmssuvc1.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐹)
45		eqid 2734	. . 3 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
46	43, 44, 45, 7	lssvscl 20904	. 2 ⊢ (((𝐹 ∈ LMod ∧ 𝐶 ∈ (LSubSp‘𝐹)) ∧ (𝑋 ∈ (Base‘(Scalar‘𝐹)) ∧ (𝑈‘𝐿) ∈ 𝐶)) → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)
47	5, 12, 19, 42, 46	syl22anc 838	1 ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 {crab 3397 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 ∅c0 4283 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 supp csupp 8100 Basecbs 17134 Scalarcsca 17178 ·_𝑠 cvsca 17179 0_gc0g 17357 Ringcrg 20166 LModclmod 20809 LSubSpclss 20880 freeLMod cfrlm 21699 unitVec cuvc 21735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-sup 9343 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-hom 17199 df-cco 17200 df-0g 17359 df-prds 17365 df-pws 17367 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-ghm 19140 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-subrg 20501 df-lmod 20811 df-lss 20881 df-lmhm 20972 df-sra 21123 df-rgmod 21124 df-dsmm 21685 df-frlm 21700 df-uvc 21736

This theorem is referenced by: (None)

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