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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 31496 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 4817 | . . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑋} ⊆ 𝑉) | |
2 | lspsnid.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lspsnid.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 2, 3 | lspssid 20962 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉) → {𝑋} ⊆ (𝑁‘{𝑋})) |
5 | 1, 4 | sylan2 591 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑋} ⊆ (𝑁‘{𝑋})) |
6 | snssg 4792 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ (𝑁‘{𝑋}) ↔ {𝑋} ⊆ (𝑁‘{𝑋}))) | |
7 | 6 | adantl 480 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ (𝑁‘{𝑋}) ↔ {𝑋} ⊆ (𝑁‘{𝑋}))) |
8 | 5, 7 | mpbird 256 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 {csn 4633 ‘cfv 6554 Basecbs 17213 LModclmod 20836 LSpanclspn 20948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-lmod 20838 df-lss 20909 df-lsp 20949 |
This theorem is referenced by: ellspsn6 20971 lssats2 20977 ellspsni 20978 lspsn 20979 lspsneq0 20989 lsmelval2 21063 lspprabs 21073 lspabs3 21102 ellspsn4 21105 lspdisjb 21107 lspfixed 21109 lindsadd 37314 lshpnelb 38682 lsateln0 38693 lssats 38710 dia1dimid 40762 dochnel 41092 dihjat1lem 41127 dochsnkr2cl 41173 lcfrvalsnN 41240 lcfrlem15 41256 mapdpglem2 41372 mapdpglem9 41379 mapdpglem12 41382 mapdpglem14 41384 mapdindp0 41418 mapdindp3 41421 hdmap11lem2 41541 hdmaprnlem3N 41549 hdmaprnlem7N 41554 hdmaprnlem8N 41555 hdmaprnlem3eN 41557 hdmaplkr 41612 |
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