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Mirrors > Home > MPE Home > Th. List > lspssp | Structured version Visualization version GIF version |
Description: If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.) |
Ref | Expression |
---|---|
lspssp.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspssp.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspssp | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lspssp.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 20814 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
4 | lspssp.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 1, 4 | lspss 20862 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ (Base‘𝑊) ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈)) |
6 | 3, 5 | syl3an2 1162 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈)) |
7 | 2, 4 | lspid 20860 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
8 | 7 | 3adant3 1130 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑈) = 𝑈) |
9 | 6, 8 | sseqtrd 4019 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ⊆ wss 3945 ‘cfv 6543 Basecbs 17174 LModclmod 20737 LSubSpclss 20809 LSpanclspn 20849 |
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 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-lmod 20739 df-lss 20810 df-lsp 20850 |
This theorem is referenced by: lspsnss 20868 lspprss 20870 lsp0 20887 lsslsp 20893 lsslspOLD 20894 lmhmlsp 20928 lspextmo 20935 lsmsp 20965 lsppratlem3 21031 lsppratlem4 21032 islbs3 21037 rspssp 21129 ocvlsp 21602 frlmsslsp 21724 ply1degltdimlem 33311 lspsslco 47496 |
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