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Mirrors > Home > MPE Home > Th. List > lspss | Structured version Visualization version GIF version |
Description: Span preserves subset ordering. (spanss 29131 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspss.v | ⊢ 𝑉 = (Base‘𝑊) |
lspss.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspss | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3 1190 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) ∧ 𝑡 ∈ (LSubSp‘𝑊)) → 𝑇 ⊆ 𝑈) | |
2 | sstr2 3922 | . . . . 5 ⊢ (𝑇 ⊆ 𝑈 → (𝑈 ⊆ 𝑡 → 𝑇 ⊆ 𝑡)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) ∧ 𝑡 ∈ (LSubSp‘𝑊)) → (𝑈 ⊆ 𝑡 → 𝑇 ⊆ 𝑡)) |
4 | 3 | ss2rabdv 4003 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡} ⊆ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑇 ⊆ 𝑡}) |
5 | intss 4859 | . . 3 ⊢ ({𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡} ⊆ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑇 ⊆ 𝑡} → ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑇 ⊆ 𝑡} ⊆ ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡}) | |
6 | 4, 5 | syl 17 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑇 ⊆ 𝑡} ⊆ ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡}) |
7 | simp1 1133 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → 𝑊 ∈ LMod) | |
8 | simp3 1135 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝑈) | |
9 | simp2 1134 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → 𝑈 ⊆ 𝑉) | |
10 | 8, 9 | sstrd 3925 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝑉) |
11 | lspss.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
12 | eqid 2798 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
13 | lspss.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
14 | 11, 12, 13 | lspval 19740 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ⊆ 𝑉) → (𝑁‘𝑇) = ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑇 ⊆ 𝑡}) |
15 | 7, 10, 14 | syl2anc 587 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) = ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑇 ⊆ 𝑡}) |
16 | 11, 12, 13 | lspval 19740 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡}) |
17 | 16 | 3adant3 1129 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑈) = ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡}) |
18 | 6, 15, 17 | 3sstr4d 3962 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 ∩ cint 4838 ‘cfv 6324 Basecbs 16475 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-lmod 19629 df-lss 19697 df-lsp 19737 |
This theorem is referenced by: lspun 19752 lspssp 19753 lspprid1 19762 lbspss 19847 lspsolvlem 19907 lspsolv 19908 lsppratlem3 19914 lbsextlem2 19924 lbsextlem3 19925 lbsextlem4 19926 lindfrn 20510 f1lindf 20511 mxidlprm 31048 idlsrgmulrss1 31064 idlsrgmulrss2 31065 lindsunlem 31108 dimkerim 31111 lindsadd 35050 lssats 36308 lpssat 36309 lssatle 36311 lssat 36312 dvhdimlem 38740 dvh3dim3N 38745 mapdindp2 39017 lspindp5 39066 |
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