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Mirrors > Home > MPE Home > Th. List > lspsnel5a | Structured version Visualization version GIF version |
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.) |
Ref | Expression |
---|---|
lspsnel5a.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspsnel5a.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspsnel5a.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspsnel5a.a | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lspsnel5a.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
Ref | Expression |
---|---|
lspsnel5a | ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnel5a.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
2 | eqid 2737 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | lspsnel5a.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | lspsnel5a.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | lspsnel5a.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | lspsnel5a.a | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
7 | 2, 3 | lssel 20350 | . . . 4 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑊)) |
8 | 6, 1, 7 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
9 | 2, 3, 4, 5, 6, 8 | lspsnel5 20408 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
10 | 1, 9 | mpbid 231 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 {csn 4584 ‘cfv 6493 Basecbs 17042 LModclmod 20274 LSubSpclss 20344 LSpanclspn 20384 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 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-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-0g 17282 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-grp 18710 df-lmod 20276 df-lss 20345 df-lsp 20385 |
This theorem is referenced by: lssats2 20413 lspsn 20415 lspsnvsi 20417 lsmelval2 20498 lspprabs 20508 lspvadd 20509 lspabs3 20534 lsmcv 20554 lspsnat 20558 lsppratlem6 20565 issubassa2 21247 lshpnel 37376 lsatel 37398 lsmsat 37401 lssatomic 37404 lssats 37405 lsat0cv 37426 dia2dimlem10 39467 dochsatshpb 39846 lclkrlem2f 39906 lcfrlem25 39961 lcfrlem35 39971 mapdval2N 40024 mapdrvallem2 40039 mapdpglem8 40073 mapdpglem13 40078 mapdindp0 40113 mapdh6aN 40129 mapdh8e 40178 mapdh9a 40183 hdmap1l6a 40203 hdmapval0 40227 hdmapval3lemN 40231 hdmap10lem 40233 hdmap11lem1 40235 hdmap11lem2 40236 hdmaprnlem4N 40247 hdmaprnlem3eN 40252 |
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