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| Mirrors > Home > MPE Home > Th. List > ellspsn5 | 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 |
|---|---|
| ellspsn5.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
| ellspsn5.n | ⊢ 𝑁 = (LSpan‘𝑊) |
| ellspsn5.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
| ellspsn5.a | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
| ellspsn5.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
| Ref | Expression |
|---|---|
| ellspsn5 | ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ellspsn5.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
| 2 | eqid 2737 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 3 | ellspsn5.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 4 | ellspsn5.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 5 | ellspsn5.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 6 | ellspsn5.a | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
| 7 | 2, 3 | lssel 20935 | . . . 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 | ellspsn5b 20993 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
| 10 | 1, 9 | mpbid 232 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 {csn 4626 ‘cfv 6561 Basecbs 17247 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-lmod 20860 df-lss 20930 df-lsp 20970 |
| This theorem is referenced by: lssats2 20998 lspsn 21000 lspsnvsi 21002 lsmelval2 21084 lspprabs 21094 lspvadd 21095 lspabs3 21123 lsmcv 21143 lspsnat 21147 lsppratlem6 21154 issubassa2 21912 lshpnel 38984 lsatel 39006 lsmsat 39009 lssatomic 39012 lssats 39013 lsat0cv 39034 dia2dimlem10 41075 dochsatshpb 41454 lclkrlem2f 41514 lcfrlem25 41569 lcfrlem35 41579 mapdval2N 41632 mapdrvallem2 41647 mapdpglem8 41681 mapdpglem13 41686 mapdindp0 41721 mapdh6aN 41737 mapdh8e 41786 mapdh9a 41791 hdmap1l6a 41811 hdmapval0 41835 hdmapval3lemN 41839 hdmap10lem 41841 hdmap11lem1 41843 hdmap11lem2 41844 hdmaprnlem4N 41855 hdmaprnlem3eN 41860 |
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