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Mirrors > Home > MPE Home > Th. List > frlmssuvc1 | Structured version Visualization version GIF version |
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.) |
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 = (0g‘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) |
Ref | Expression |
---|---|
frlmssuvc1 | ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶) |
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 20438 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ LMod) |
5 | 1, 2, 4 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐹 ∈ LMod) |
6 | frlmssuvc1.j | . . 3 ⊢ (𝜑 → 𝐽 ⊆ 𝐼) | |
7 | eqid 2798 | . . . 4 ⊢ (LSubSp‘𝐹) = (LSubSp‘𝐹) | |
8 | frlmssuvc1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐹) | |
9 | frlmssuvc1.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
10 | frlmssuvc1.c | . . . 4 ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} | |
11 | 3, 7, 8, 9, 10 | frlmsslss2 20464 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ (LSubSp‘𝐹)) |
12 | 1, 2, 6, 11 | syl3anc 1368 | . 2 ⊢ (𝜑 → 𝐶 ∈ (LSubSp‘𝐹)) |
13 | frlmssuvc1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
14 | frlmssuvc1.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
15 | 3 | frlmsca 20442 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝐹)) |
16 | 1, 2, 15 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) |
17 | 16 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝐹))) |
18 | 14, 17 | syl5eq 2845 | . . 3 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝐹))) |
19 | 13, 18 | eleqtrd 2892 | . 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝐹))) |
20 | frlmssuvc1.u | . . . . . 6 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
21 | 20, 3, 8 | uvcff 20480 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶𝐵) |
22 | 1, 2, 21 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝑈:𝐼⟶𝐵) |
23 | frlmssuvc1.l | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ 𝐽) | |
24 | 6, 23 | sseldd 3916 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ 𝐼) |
25 | 22, 24 | ffvelrnd 6829 | . . 3 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐵) |
26 | 3, 14, 8 | frlmbasf 20449 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑈‘𝐿) ∈ 𝐵) → (𝑈‘𝐿):𝐼⟶𝐾) |
27 | 2, 25, 26 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝑈‘𝐿):𝐼⟶𝐾) |
28 | 1 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑅 ∈ Ring) |
29 | 2 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐼 ∈ 𝑉) |
30 | 24 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐿 ∈ 𝐼) |
31 | eldifi 4054 | . . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼) | |
32 | 31 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼) |
33 | disjdif 4379 | . . . . . 6 ⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ | |
34 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ (𝐼 ∖ 𝐽)) | |
35 | disjne 4362 | . . . . . 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 20479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝐿)‘𝑥) = 0 ) |
38 | 27, 37 | suppss 7843 | . . 3 ⊢ (𝜑 → ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽) |
39 | oveq1 7142 | . . . . 5 ⊢ (𝑥 = (𝑈‘𝐿) → (𝑥 supp 0 ) = ((𝑈‘𝐿) supp 0 )) | |
40 | 39 | sseq1d 3946 | . . . 4 ⊢ (𝑥 = (𝑈‘𝐿) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽)) |
41 | 40, 10 | elrab2 3631 | . . 3 ⊢ ((𝑈‘𝐿) ∈ 𝐶 ↔ ((𝑈‘𝐿) ∈ 𝐵 ∧ ((𝑈‘𝐿) supp 0 ) ⊆ 𝐽)) |
42 | 25, 38, 41 | sylanbrc 586 | . 2 ⊢ (𝜑 → (𝑈‘𝐿) ∈ 𝐶) |
43 | eqid 2798 | . . 3 ⊢ (Scalar‘𝐹) = (Scalar‘𝐹) | |
44 | frlmssuvc1.t | . . 3 ⊢ · = ( ·𝑠 ‘𝐹) | |
45 | eqid 2798 | . . 3 ⊢ (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹)) | |
46 | 43, 44, 45, 7 | lssvscl 19720 | . 2 ⊢ (((𝐹 ∈ LMod ∧ 𝐶 ∈ (LSubSp‘𝐹)) ∧ (𝑋 ∈ (Base‘(Scalar‘𝐹)) ∧ (𝑈‘𝐿) ∈ 𝐶)) → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶) |
47 | 5, 12, 19, 42, 46 | syl22anc 837 | 1 ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supp csupp 7813 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 Ringcrg 19290 LModclmod 19627 LSubSpclss 19696 freeLMod cfrlm 20435 unitVec cuvc 20471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-ghm 18348 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lmhm 19787 df-sra 19937 df-rgmod 19938 df-dsmm 20421 df-frlm 20436 df-uvc 20472 |
This theorem is referenced by: (None) |
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