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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 2610 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lspssp.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 18758 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
4 | lspssp.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 1, 4 | lspss 18805 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ (Base‘𝑊) ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈)) |
6 | 3, 5 | syl3an2 1352 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈)) |
7 | 2, 4 | lspid 18803 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
8 | 7 | 3adant3 1074 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑈) = 𝑈) |
9 | 6, 8 | sseqtrd 3604 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 Basecbs 15695 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-lmod 18688 df-lss 18754 df-lsp 18793 |
This theorem is referenced by: lspsnss 18811 lspprss 18813 lsp0 18830 lsslsp 18836 lmhmlsp 18870 lspextmo 18877 lsmsp 18907 lsppratlem3 18970 lsppratlem4 18971 islbs3 18976 rspssp 19047 ocvlsp 19839 frlmsslsp 19954 lspsslco 42020 |
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