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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 2623 | . 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
2 | eqidd 2623 | . 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 2623 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
6 | eqidd 2623 | . 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 3624 | . . 3 ⊢ 𝑉 ⊆ 𝑉 | |
10 | 9 | a1i 11 | . 2 ⊢ (𝑊 ∈ LMod → 𝑉 ⊆ 𝑉) |
11 | 3 | lmodbn0 18873 | . 2 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅) |
12 | simpl 473 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑊 ∈ LMod) | |
13 | eqid 2622 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
14 | eqid 2622 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
15 | eqid 2622 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
16 | 3, 13, 14, 15 | lmodvscl 18880 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉) |
17 | 16 | 3adant3r3 1276 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉) |
18 | simpr3 1069 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
19 | eqid 2622 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
20 | 3, 19 | lmodvacl 18877 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑉) |
21 | 12, 17, 18, 20 | syl3anc 1326 | . 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 18936 | 1 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Scalarcsca 15944 ·𝑠 cvsca 15945 LModclmod 18863 LSubSpclss 18932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-lmod 18865 df-lss 18933 |
This theorem is referenced by: lssuni 18940 islss3 18959 lssmre 18966 lspf 18974 lspval 18975 lmhmrnlss 19050 lidl1 19220 aspval 19328 isphld 19999 ocv1 20023 islshpcv 34340 dochexmidlem8 36756 hdmaprnlem4N 37145 lnmfg 37652 |
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