frlmssuvc1 - Metamath Proof Explorer

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Theorem frlmssuvc1 21011

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 20966	. . 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 2738	. . . 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 20992	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ (LSubSp‘𝐹))
12	1, 2, 6, 11	syl3anc 1370	. 2 ⊢ (𝜑 → 𝐶 ∈ (LSubSp‘𝐹))
13		frlmssuvc1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
14		frlmssuvc1.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
15	3	frlmsca 20970	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹))
16	1, 2, 15	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))
17	16	fveq2d 6770	. . . 4 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
18	14, 17	eqtrid 2790	. . 3 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝐹)))
19	13, 18	eleqtrd 2841	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝐹)))
20		frlmssuvc1.u	. . . . . 6 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
21	20, 3, 8	uvcff 21008	. . . . 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 3921	. . . 4 ⊢ (𝜑 → 𝐿 ∈ 𝐼)
25	22, 24	ffvelrnd 6954	. . 3 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐵)
26	3, 14, 8	frlmbasf 20977	. . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑅 ∈ Ring)
29	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐼 ∈ 𝑉)
30	24	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ∈ 𝐼)
31		eldifi 4060	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼)
32	31	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼)
33		disjdif 4405	. . . . . 6 ⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅
34		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ (𝐼 ∖ 𝐽))
35		disjne 4388	. . . . . 6 ⊢ (((𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ ∧ 𝐿 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ≠ 𝑥)
36	33, 23, 34, 35	mp3an2ani 1467	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ≠ 𝑥)
37	20, 28, 29, 30, 32, 36, 9	uvcvv0 21007	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝐿)‘𝑥) = 0 )
38	27, 37	suppss 7997	. . 3 ⊢ (𝜑 → ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽)
39		oveq1 7274	. . . . 5 ⊢ (𝑥 = (𝑈‘𝐿) → (𝑥 supp 0 ) = ((𝑈‘𝐿) supp 0 ))
40	39	sseq1d 3951	. . . 4 ⊢ (𝑥 = (𝑈‘𝐿) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽))
41	40, 10	elrab2 3626	. . 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 2738	. . 3 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹)
44		frlmssuvc1.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐹)
45		eqid 2738	. . 3 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
46	43, 44, 45, 7	lssvscl 20227	. 2 ⊢ (((𝐹 ∈ LMod ∧ 𝐶 ∈ (LSubSp‘𝐹)) ∧ (𝑋 ∈ (Base‘(Scalar‘𝐹)) ∧ (𝑈‘𝐿) ∈ 𝐶)) → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)
47	5, 12, 19, 42, 46	syl22anc 836	1 ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 {crab 3068 ∖ cdif 3883 ∩ cin 3885 ⊆ wss 3886 ∅c0 4256 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 supp csupp 7964 Basecbs 16922 Scalarcsca 16975 ·_𝑠 cvsca 16976 0_gc0g 17160 Ringcrg 19793 LModclmod 20133 LSubSpclss 20203 freeLMod cfrlm 20963 unitVec cuvc 20999

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-sup 9188 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-fz 13250 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-hom 16996 df-cco 16997 df-0g 17162 df-prds 17168 df-pws 17170 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-mhm 18440 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-subg 18762 df-ghm 18842 df-mgp 19731 df-ur 19748 df-ring 19795 df-subrg 20032 df-lmod 20135 df-lss 20204 df-lmhm 20294 df-sra 20444 df-rgmod 20445 df-dsmm 20949 df-frlm 20964 df-uvc 21000

This theorem is referenced by: (None)

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