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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 2611 | . 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
2 | eqidd 2611 | . 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 2611 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
6 | eqidd 2611 | . 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 | ssid 3587 | . . 3 ⊢ 𝑉 ⊆ 𝑉 | |
10 | 9 | a1i 11 | . 2 ⊢ (𝑊 ∈ LMod → 𝑉 ⊆ 𝑉) |
11 | 3 | lmodbn0 18696 | . 2 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅) |
12 | simpl 472 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑊 ∈ LMod) | |
13 | eqid 2610 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
14 | eqid 2610 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
15 | eqid 2610 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
16 | 3, 13, 14, 15 | lmodvscl 18703 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉) |
17 | 16 | 3adant3r3 1268 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉) |
18 | simpr3 1062 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
19 | eqid 2610 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
20 | 3, 19 | lmodvacl 18700 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑉) |
21 | 12, 17, 18, 20 | syl3anc 1318 | . 2 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑉) |
22 | 1, 2, 4, 5, 6, 8, 10, 11, 21 | islssd 18757 | 1 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 LModclmod 18686 LSubSpclss 18753 |
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-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-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-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-iota 5768 df-fun 5806 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 |
This theorem is referenced by: lssuni 18761 islss3 18780 lssmre 18787 lspf 18795 lspval 18796 lmhmrnlss 18871 lidl1 19041 aspval 19149 isphld 19818 ocv1 19842 islshpcv 33358 dochexmidlem8 35774 hdmaprnlem4N 36163 lnmfg 36670 |
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