Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2610 | . . 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 18759 | . . . 4 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑊)) |
8 | 6, 1, 7 | syl2anc 691 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
9 | 2, 3, 4, 5, 6, 8 | lspsnel5 18816 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
10 | 1, 9 | mpbid 221 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 ‘cfv 5804 Basecbs 15695 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-lmod 18688 df-lss 18754 df-lsp 18793 |
This theorem is referenced by: lssats2 18821 lspsn 18823 lspsnvsi 18825 lsmelval2 18906 lspprabs 18916 lspvadd 18917 lspabs3 18942 lsmcv 18962 lspsnat 18966 lsppratlem6 18973 issubassa2 19166 lshpnel 33288 lsatel 33310 lsmsat 33313 lssatomic 33316 lssats 33317 lsat0cv 33338 dia2dimlem10 35380 dochsatshpb 35759 lclkrlem2f 35819 lcfrlem25 35874 lcfrlem35 35884 mapdval2N 35937 mapdrvallem2 35952 mapdpglem8 35986 mapdpglem13 35991 mapdindp0 36026 mapdh6aN 36042 mapdh8e 36091 mapdh9a 36097 hdmap1l6a 36117 hdmapval0 36143 hdmapval3lemN 36147 hdmap10lem 36149 hdmap11lem1 36151 hdmap11lem2 36152 hdmaprnlem4N 36163 hdmaprnlem3eN 36168 |
Copyright terms: Public domain | W3C validator |