frlmssuvc1 - Metamath Proof Explorer

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Theorem frlmssuvc1 21996

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 21951	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ LMod)
5	1, 2, 4	syl2anc 596	. 2 ⊢ (𝜑 → 𝐹 ∈ LMod)
6		frlmssuvc1.j	. . 3 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
7		eqid 2765	. . . 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 21977	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ (LSubSp‘𝐹))
12	1, 2, 6, 11	syl3anc 1398	. 2 ⊢ (𝜑 → 𝐶 ∈ (LSubSp‘𝐹))
13		frlmssuvc1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
14		frlmssuvc1.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
15	3	frlmsca 21955	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
16	1, 2, 15	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
17	16	fveq2d 6889	. . . 4 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
18	14, 17	eqtrid 2812	. . 3 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝐹)))
19	13, 18	eleqtrd 2867	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝐹)))
20		frlmssuvc1.u	. . . . . 6 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
21	20, 3, 8	uvcff 21993	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶𝐵)
22	1, 2, 21	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝑈:𝐼⟶𝐵)
23		frlmssuvc1.l	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ 𝐽)
24	6, 23	sseldd 3939	. . . 4 ⊢ (𝜑 → 𝐿 ∈ 𝐼)
25	22, 24	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐵)
26	3, 14, 8	frlmbasf 21962	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑈‘𝐿) ∈ 𝐵) → (𝑈‘𝐿):𝐼⟶𝐾)
27	2, 25, 26	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑈‘𝐿):𝐼⟶𝐾)
28	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑅 ∈ Ring)
29	2	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐼 ∈ 𝑉)
30	24	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ∈ 𝐼)
31		eldifi 4085	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼)
32	31	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼)
33		disjdif 4433	. . . . . 6 ⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅
34		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ (𝐼 ∖ 𝐽))
35		disjne 4415	. . . . . 6 ⊢ (((𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ ∧ 𝐿 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ≠ 𝑥)
36	33, 23, 34, 35	mp3an2ani 1497	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ≠ 𝑥)
37	20, 28, 29, 30, 32, 36, 9	uvcvv0 21992	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝐿)‘𝑥) = 0 )
38	27, 37	suppss 8196	. . 3 ⊢ (𝜑 → ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽)
39		oveq1 7426	. . . . 5 ⊢ (𝑥 = (𝑈‘𝐿) → (𝑥 supp 0 ) = ((𝑈‘𝐿) supp 0 ))
40	39	sseq1d 3969	. . . 4 ⊢ (𝑥 = (𝑈‘𝐿) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽))
41	40, 10	elrab2 3656	. . 3 ⊢ ((𝑈‘𝐿) ∈ 𝐶 ↔ ((𝑈‘𝐿) ∈ 𝐵 ∧ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽))
42	25, 38, 41	sylanbrc 595	. 2 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐶)
43		eqid 2765	. . 3 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
44		frlmssuvc1.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐹)
45		eqid 2765	. . 3 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
46	43, 44, 45, 7	lssvscl 21128	. 2 ⊢ (((𝐹 ∈ LMod ∧ 𝐶 ∈ (LSubSp‘𝐹)) ∧ (𝑋 ∈ (Base‘(Scalar‘𝐹)) ∧ (𝑈‘𝐿) ∈ 𝐶)) → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)
47	5, 12, 19, 42, 46	syl22anc 852	1 ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 {crab 3418 ∖ cdif 3903 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 supp csupp 8162 Basecbs 17293 Scalarcsca 17337 ·_𝑠 cvsca 17338 0_gc0g 17516 Ringcrg 20361 LModclmod 21033 LSubSpclss 21104 freeLMod cfrlm 21948 unitVec cuvc 21984

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-sup 9409 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-fz 13554 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ds 17356 df-hom 17358 df-cco 17359 df-0g 17518 df-prds 17524 df-pws 17526 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-mhm 18880 df-submnd 18881 df-grp 19049 df-minusg 19050 df-sbg 19051 df-subg 19235 df-ghm 19330 df-cmn 19898 df-abl 19899 df-mgp 20263 df-rng 20277 df-ur 20310 df-ring 20363 df-subrg 20721 df-lmod 21035 df-lss 21105 df-lmhm 21195 df-sra 21346 df-rgmod 21347 df-dsmm 21934 df-frlm 21949 df-uvc 21985

This theorem is used by: (None)

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