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| Mirrors > Home > MPE Home > Th. List > lspsnid | Structured version Visualization version GIF version | ||
| Description: A vector belongs to the span of its singleton. (spansnid 31582 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| lspsnid.v | ⊢ 𝑉 = (Base‘𝑊) |
| lspsnid.n | ⊢ 𝑁 = (LSpan‘𝑊) |
| Ref | Expression |
|---|---|
| lspsnid | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snssi 4808 | . . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑋} ⊆ 𝑉) | |
| 2 | lspsnid.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
| 3 | lspsnid.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 4 | 2, 3 | lspssid 20983 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉) → {𝑋} ⊆ (𝑁‘{𝑋})) |
| 5 | 1, 4 | sylan2 593 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑋} ⊆ (𝑁‘{𝑋})) |
| 6 | snssg 4783 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ (𝑁‘{𝑋}) ↔ {𝑋} ⊆ (𝑁‘{𝑋}))) | |
| 7 | 6 | adantl 481 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ (𝑁‘{𝑋}) ↔ {𝑋} ⊆ (𝑁‘{𝑋}))) |
| 8 | 5, 7 | mpbird 257 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 {csn 4626 ‘cfv 6561 Basecbs 17247 LModclmod 20858 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: ellspsn6 20992 lssats2 20998 ellspsni 20999 lspsn 21000 lspsneq0 21010 lsmelval2 21084 lspprabs 21094 lspabs3 21123 ellspsn4 21126 lspdisjb 21128 lspfixed 21130 lindsadd 37620 lshpnelb 38985 lsateln0 38996 lssats 39013 dia1dimid 41065 dochnel 41395 dihjat1lem 41430 dochsnkr2cl 41476 lcfrvalsnN 41543 lcfrlem15 41559 mapdpglem2 41675 mapdpglem9 41682 mapdpglem12 41685 mapdpglem14 41687 mapdindp0 41721 mapdindp3 41724 hdmap11lem2 41844 hdmaprnlem3N 41852 hdmaprnlem7N 41857 hdmaprnlem8N 41858 hdmaprnlem3eN 41860 hdmaplkr 41915 |
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