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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 2735 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lspssp.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 20952 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
4 | lspssp.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 1, 4 | lspss 21000 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ (Base‘𝑊) ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈)) |
6 | 3, 5 | syl3an2 1163 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈)) |
7 | 2, 4 | lspid 20998 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
8 | 7 | 3adant3 1131 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑈) = 𝑈) |
9 | 6, 8 | sseqtrd 4036 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 Basecbs 17245 LModclmod 20875 LSubSpclss 20947 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: lspsnss 21006 lspprss 21008 lsp0 21025 lsslsp 21031 lsslspOLD 21032 lmhmlsp 21066 lspextmo 21073 lsmsp 21103 lsppratlem3 21169 lsppratlem4 21170 islbs3 21175 rspssp 21267 ocvlsp 21712 frlmsslsp 21834 ply1degltdimlem 33650 lspsslco 48283 |
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