frlmssuvc1 - Metamath Proof Explorer

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Theorem frlmssuvc1 21727

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 21682	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ LMod)
5	1, 2, 4	syl2anc 583	. 2 ⊢ (𝜑 → 𝐹 ∈ LMod)
6		frlmssuvc1.j	. . 3 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
7		eqid 2728	. . . 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 21708	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ (LSubSp‘𝐹))
12	1, 2, 6, 11	syl3anc 1369	. 2 ⊢ (𝜑 → 𝐶 ∈ (LSubSp‘𝐹))
13		frlmssuvc1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
14		frlmssuvc1.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
15	3	frlmsca 21686	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
16	1, 2, 15	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
17	16	fveq2d 6901	. . . 4 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
18	14, 17	eqtrid 2780	. . 3 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝐹)))
19	13, 18	eleqtrd 2831	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝐹)))
20		frlmssuvc1.u	. . . . . 6 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
21	20, 3, 8	uvcff 21724	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶𝐵)
22	1, 2, 21	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝑈:𝐼⟶𝐵)
23		frlmssuvc1.l	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ 𝐽)
24	6, 23	sseldd 3981	. . . 4 ⊢ (𝜑 → 𝐿 ∈ 𝐼)
25	22, 24	ffvelcdmd 7095	. . 3 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐵)
26	3, 14, 8	frlmbasf 21693	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑈‘𝐿) ∈ 𝐵) → (𝑈‘𝐿):𝐼⟶𝐾)
27	2, 25, 26	syl2anc 583	. . . 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 4125	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼)
32	31	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼)
33		disjdif 4472	. . . . . 6 ⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅
34		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ (𝐼 ∖ 𝐽))
35		disjne 4455	. . . . . 6 ⊢ (((𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ ∧ 𝐿 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ≠ 𝑥)
36	33, 23, 34, 35	mp3an2ani 1465	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ≠ 𝑥)
37	20, 28, 29, 30, 32, 36, 9	uvcvv0 21723	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝐿)‘𝑥) = 0 )
38	27, 37	suppss 8198	. . 3 ⊢ (𝜑 → ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽)
39		oveq1 7427	. . . . 5 ⊢ (𝑥 = (𝑈‘𝐿) → (𝑥 supp 0 ) = ((𝑈‘𝐿) supp 0 ))
40	39	sseq1d 4011	. . . 4 ⊢ (𝑥 = (𝑈‘𝐿) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽))
41	40, 10	elrab2 3685	. . 3 ⊢ ((𝑈‘𝐿) ∈ 𝐶 ↔ ((𝑈‘𝐿) ∈ 𝐵 ∧ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽))
42	25, 38, 41	sylanbrc 582	. 2 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐶)
43		eqid 2728	. . 3 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
44		frlmssuvc1.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐹)
45		eqid 2728	. . 3 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
46	43, 44, 45, 7	lssvscl 20838	. 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 1534 ∈ wcel 2099 ≠ wne 2937 {crab 3429 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 ∅c0 4323 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 supp csupp 8165 Basecbs 17179 Scalarcsca 17235 ·_𝑠 cvsca 17236 0_gc0g 17420 Ringcrg 20172 LModclmod 20742 LSubSpclss 20814 freeLMod cfrlm 21679 unitVec cuvc 21715

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-prds 17428 df-pws 17430 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-ghm 19167 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-subrg 20507 df-lmod 20744 df-lss 20815 df-lmhm 20906 df-sra 21057 df-rgmod 21058 df-dsmm 21665 df-frlm 21680 df-uvc 21716

This theorem is referenced by: (None)

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