frlmssuvc1 - Metamath Proof Explorer

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Theorem frlmssuvc1 21679

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 21634	. . 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 2729	. . . 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 21660	. . 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 21638	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
16	1, 2, 15	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
17	16	fveq2d 6844	. . . 4 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
18	14, 17	eqtrid 2776	. . 3 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝐹)))
19	13, 18	eleqtrd 2830	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝐹)))
20		frlmssuvc1.u	. . . . . 6 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
21	20, 3, 8	uvcff 21676	. . . . 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 3944	. . . 4 ⊢ (𝜑 → 𝐿 ∈ 𝐼)
25	22, 24	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐵)
26	3, 14, 8	frlmbasf 21645	. . . . 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 4090	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼)
32	31	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼)
33		disjdif 4431	. . . . . 6 ⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅
34		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ (𝐼 ∖ 𝐽))
35		disjne 4414	. . . . . 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 21675	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝐿)‘𝑥) = 0 )
38	27, 37	suppss 8150	. . 3 ⊢ (𝜑 → ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽)
39		oveq1 7376	. . . . 5 ⊢ (𝑥 = (𝑈‘𝐿) → (𝑥 supp 0 ) = ((𝑈‘𝐿) supp 0 ))
40	39	sseq1d 3975	. . . 4 ⊢ (𝑥 = (𝑈‘𝐿) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽))
41	40, 10	elrab2 3659	. . 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 2729	. . 3 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
44		frlmssuvc1.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐹)
45		eqid 2729	. . 3 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
46	43, 44, 45, 7	lssvscl 20837	. 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 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3402 ∖ cdif 3908 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 supp csupp 8116 Basecbs 17155 Scalarcsca 17199 ·_𝑠 cvsca 17200 0_gc0g 17378 Ringcrg 20118 LModclmod 20742 LSubSpclss 20813 freeLMod cfrlm 21631 unitVec cuvc 21667

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17380 df-prds 17386 df-pws 17388 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-mhm 18686 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-subg 19031 df-ghm 19121 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-subrg 20455 df-lmod 20744 df-lss 20814 df-lmhm 20905 df-sra 21056 df-rgmod 21057 df-dsmm 21617 df-frlm 21632 df-uvc 21668

This theorem is referenced by: (None)

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