![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lspsnid | Structured version Visualization version GIF version |
Description: A vector belongs to the span of its singleton. (spansnid 31592 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 4813 | . . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑋} ⊆ 𝑉) | |
2 | lspsnid.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lspsnid.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 2, 3 | lspssid 21001 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉) → {𝑋} ⊆ (𝑁‘{𝑋})) |
5 | 1, 4 | sylan2 593 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑋} ⊆ (𝑁‘{𝑋})) |
6 | snssg 4788 | . . 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 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 ‘cfv 6563 Basecbs 17245 LModclmod 20875 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: ellspsn6 21010 lssats2 21016 ellspsni 21017 lspsn 21018 lspsneq0 21028 lsmelval2 21102 lspprabs 21112 lspabs3 21141 ellspsn4 21144 lspdisjb 21146 lspfixed 21148 lindsadd 37600 lshpnelb 38966 lsateln0 38977 lssats 38994 dia1dimid 41046 dochnel 41376 dihjat1lem 41411 dochsnkr2cl 41457 lcfrvalsnN 41524 lcfrlem15 41540 mapdpglem2 41656 mapdpglem9 41663 mapdpglem12 41666 mapdpglem14 41668 mapdindp0 41702 mapdindp3 41705 hdmap11lem2 41825 hdmaprnlem3N 41833 hdmaprnlem7N 41838 hdmaprnlem8N 41839 hdmaprnlem3eN 41841 hdmaplkr 41896 |
Copyright terms: Public domain | W3C validator |