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Mirrors > Home > MPE Home > Th. List > lss1 | Structured version Visualization version GIF version |
Description: The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lssss.v | ⊢ 𝑉 = (Base‘𝑊) |
lssss.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lss1 | ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2736 | . 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
2 | eqidd 2736 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))) | |
3 | lssss.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
4 | 3 | a1i 11 | . 2 ⊢ (𝑊 ∈ LMod → 𝑉 = (Base‘𝑊)) |
5 | eqidd 2736 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
6 | eqidd 2736 | . 2 ⊢ (𝑊 ∈ LMod → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)) | |
7 | lssss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
8 | 7 | a1i 11 | . 2 ⊢ (𝑊 ∈ LMod → 𝑆 = (LSubSp‘𝑊)) |
9 | ssidd 4019 | . 2 ⊢ (𝑊 ∈ LMod → 𝑉 ⊆ 𝑉) | |
10 | 3 | lmodbn0 20886 | . 2 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅) |
11 | simpl 482 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑊 ∈ LMod) | |
12 | eqid 2735 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
13 | eqid 2735 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
14 | eqid 2735 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
15 | 3, 12, 13, 14 | lmodvscl 20893 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉) |
16 | 15 | 3adant3r3 1183 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉) |
17 | simpr3 1195 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
18 | eqid 2735 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
19 | 3, 18 | lmodvacl 20890 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑉) |
20 | 11, 16, 17, 19 | syl3anc 1370 | . 2 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑉) |
21 | 1, 2, 4, 5, 6, 8, 9, 10, 20 | islssd 20951 | 1 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 LModclmod 20875 LSubSpclss 20947 |
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-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-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-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-iota 6516 df-fun 6565 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 |
This theorem is referenced by: lssuni 20955 islss3 20975 lssmre 20982 lspf 20990 lspval 20991 lmhmrnlss 21067 lidl1ALT 21259 isphld 21690 ocv1 21715 aspval 21911 islshpcv 39035 dochexmidlem8 41450 hdmaprnlem4N 41836 lnmfg 43071 |
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