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| Mirrors > Home > MPE Home > Th. List > ellspsn5b | Structured version Visualization version GIF version | ||
| Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) | 
| Ref | Expression | 
|---|---|
| ellspsn5b.v | ⊢ 𝑉 = (Base‘𝑊) | 
| ellspsn5b.s | ⊢ 𝑆 = (LSubSp‘𝑊) | 
| ellspsn5b.n | ⊢ 𝑁 = (LSpan‘𝑊) | 
| ellspsn5b.w | ⊢ (𝜑 → 𝑊 ∈ LMod) | 
| ellspsn5b.a | ⊢ (𝜑 → 𝑈 ∈ 𝑆) | 
| ellspsn5b.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) | 
| Ref | Expression | 
|---|---|
| ellspsn5b | ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ellspsn5b.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 2 | ellspsn5b.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
| 3 | ellspsn5b.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 4 | ellspsn5b.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 5 | ellspsn5b.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 6 | ellspsn5b.a | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
| 7 | 2, 3, 4, 5, 6 | ellspsn6 20993 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) | 
| 8 | 1, 7 | mpbirand 707 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 {csn 4625 ‘cfv 6560 Basecbs 17248 LModclmod 20859 LSubSpclss 20930 LSpanclspn 20970 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-lmod 20861 df-lss 20931 df-lsp 20971 | 
| This theorem is referenced by: ellspsn5 20995 lspprid1 20996 lspsnss2 21004 lsmelpr 21091 lspsncmp 21119 lspsnne1 21120 lspsnne2 21121 lspsneq 21125 lspindpi 21135 islbs2 21157 lindsadd 37621 lindsenlbs 37623 lsatelbN 39008 lsmsat 39010 lsatfixedN 39011 l1cvpat 39056 dia2dimlem5 41071 dochsncom 41385 dihjat1lem 41431 dvh4dimlem 41446 lclkrlem2a 41510 lcfrlem6 41550 lcfrlem20 41565 lcfrlem26 41571 lcfrlem36 41581 mapdval2N 41633 mapdrvallem2 41648 mapdindp 41674 mapdh6aN 41738 lspindp5 41773 mapdh8ab 41780 mapdh8e 41787 hdmap1l6a 41812 hdmaprnlem3eN 41861 hdmapoc 41934 | 
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