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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 21010 | . 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 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 ‘cfv 6563 Basecbs 17245 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-lmod 20877 df-lss 20948 df-lsp 20988 |
This theorem is referenced by: ellspsn5 21012 lspprid1 21013 lspsnss2 21021 lsmelpr 21108 lspsncmp 21136 lspsnne1 21137 lspsnne2 21138 lspsneq 21142 lspindpi 21152 islbs2 21174 lindsadd 37600 lindsenlbs 37602 lsatelbN 38988 lsmsat 38990 lsatfixedN 38991 l1cvpat 39036 dia2dimlem5 41051 dochsncom 41365 dihjat1lem 41411 dvh4dimlem 41426 lclkrlem2a 41490 lcfrlem6 41530 lcfrlem20 41545 lcfrlem26 41551 lcfrlem36 41561 mapdval2N 41613 mapdrvallem2 41628 mapdindp 41654 mapdh6aN 41718 lspindp5 41753 mapdh8ab 41760 mapdh8e 41767 hdmap1l6a 41792 hdmaprnlem3eN 41841 hdmapoc 41914 |
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