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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 2738 | . 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
| 2 | eqidd 2738 | . 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 2738 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
| 6 | eqidd 2738 | . 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 4007 | . 2 ⊢ (𝑊 ∈ LMod → 𝑉 ⊆ 𝑉) | |
| 10 | 3 | lmodbn0 20869 | . 2 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅) | 
| 11 | simpl 482 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑊 ∈ LMod) | |
| 12 | eqid 2737 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 13 | eqid 2737 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 14 | eqid 2737 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 15 | 3, 12, 13, 14 | lmodvscl 20876 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉) | 
| 16 | 15 | 3adant3r3 1185 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉) | 
| 17 | simpr3 1197 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
| 18 | eqid 2737 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 19 | 3, 18 | lmodvacl 20873 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑉) | 
| 20 | 11, 16, 17, 19 | syl3anc 1373 | . 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 20933 | 1 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 LModclmod 20858 LSubSpclss 20929 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-lmod 20860 df-lss 20930 | 
| This theorem is referenced by: lssuni 20937 islss3 20957 lssmre 20964 lspf 20972 lspval 20973 lmhmrnlss 21049 lidl1ALT 21241 isphld 21672 ocv1 21697 aspval 21893 islshpcv 39054 dochexmidlem8 41469 hdmaprnlem4N 41855 lnmfg 43094 | 
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