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Mirrors > Home > MPE Home > Th. List > lsssn0 | Structured version Visualization version GIF version |
Description: The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lss0cl.z | ⊢ 0 = (0g‘𝑊) |
lss0cl.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lsssn0 | ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2779 | . 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
2 | eqidd 2779 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))) | |
3 | eqidd 2779 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) = (Base‘𝑊)) | |
4 | eqidd 2779 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
5 | eqidd 2779 | . 2 ⊢ (𝑊 ∈ LMod → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)) | |
6 | lss0cl.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 6 | a1i 11 | . 2 ⊢ (𝑊 ∈ LMod → 𝑆 = (LSubSp‘𝑊)) |
8 | eqid 2778 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
9 | lss0cl.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
10 | 8, 9 | lmod0vcl 19284 | . . 3 ⊢ (𝑊 ∈ LMod → 0 ∈ (Base‘𝑊)) |
11 | 10 | snssd 4571 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ⊆ (Base‘𝑊)) |
12 | 9 | fvexi 6460 | . . . 4 ⊢ 0 ∈ V |
13 | 12 | snnz 4542 | . . 3 ⊢ { 0 } ≠ ∅ |
14 | 13 | a1i 11 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ≠ ∅) |
15 | simpr2 1207 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑎 ∈ { 0 }) | |
16 | elsni 4415 | . . . . . . . 8 ⊢ (𝑎 ∈ { 0 } → 𝑎 = 0 ) | |
17 | 15, 16 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑎 = 0 ) |
18 | 17 | oveq2d 6938 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊)𝑎) = (𝑥( ·𝑠 ‘𝑊) 0 )) |
19 | eqid 2778 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
20 | eqid 2778 | . . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
21 | eqid 2778 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
22 | 19, 20, 21, 9 | lmodvs0 19289 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝑥( ·𝑠 ‘𝑊) 0 ) = 0 ) |
23 | 22 | 3ad2antr1 1196 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊) 0 ) = 0 ) |
24 | 18, 23 | eqtrd 2814 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊)𝑎) = 0 ) |
25 | simpr3 1209 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑏 ∈ { 0 }) | |
26 | elsni 4415 | . . . . . 6 ⊢ (𝑏 ∈ { 0 } → 𝑏 = 0 ) | |
27 | 25, 26 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑏 = 0 ) |
28 | 24, 27 | oveq12d 6940 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = ( 0 (+g‘𝑊) 0 )) |
29 | eqid 2778 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
30 | 8, 29, 9 | lmod0vlid 19285 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 0 ∈ (Base‘𝑊)) → ( 0 (+g‘𝑊) 0 ) = 0 ) |
31 | 10, 30 | mpdan 677 | . . . . 5 ⊢ (𝑊 ∈ LMod → ( 0 (+g‘𝑊) 0 ) = 0 ) |
32 | 31 | adantr 474 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ( 0 (+g‘𝑊) 0 ) = 0 ) |
33 | 28, 32 | eqtrd 2814 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
34 | ovex 6954 | . . . 4 ⊢ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ V | |
35 | 34 | elsn 4413 | . . 3 ⊢ (((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ { 0 } ↔ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
36 | 33, 35 | sylibr 226 | . 2 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ { 0 }) |
37 | 1, 2, 3, 4, 5, 7, 11, 14, 36 | islssd 19328 | 1 ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∅c0 4141 {csn 4398 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 Scalarcsca 16341 ·𝑠 cvsca 16342 0gc0g 16486 LModclmod 19255 LSubSpclss 19324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-mgp 18877 df-ring 18936 df-lmod 19257 df-lss 19325 |
This theorem is referenced by: lspsn0 19403 lsp0 19404 lmhmkerlss 19446 lidl0 19616 lsatcv0 35187 lsatcveq0 35188 lsat0cv 35189 lsatcv0eq 35203 dochsat 37539 mapd0 37821 mapdcnvatN 37822 mapdat 37823 mapdn0 37825 hdmapeq0 38000 |
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