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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 2726 | . 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
2 | eqidd 2726 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))) | |
3 | eqidd 2726 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) = (Base‘𝑊)) | |
4 | eqidd 2726 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
5 | eqidd 2726 | . 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 2725 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
9 | lss0cl.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
10 | 8, 9 | lmod0vcl 20814 | . . 3 ⊢ (𝑊 ∈ LMod → 0 ∈ (Base‘𝑊)) |
11 | 10 | snssd 4817 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ⊆ (Base‘𝑊)) |
12 | 9 | fvexi 6914 | . . . 4 ⊢ 0 ∈ V |
13 | 12 | snnz 4784 | . . 3 ⊢ { 0 } ≠ ∅ |
14 | 13 | a1i 11 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ≠ ∅) |
15 | simpr2 1192 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑎 ∈ { 0 }) | |
16 | elsni 4649 | . . . . . . . 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 7439 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊)𝑎) = (𝑥( ·𝑠 ‘𝑊) 0 )) |
19 | eqid 2725 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
20 | eqid 2725 | . . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
21 | eqid 2725 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
22 | 19, 20, 21, 9 | lmodvs0 20819 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝑥( ·𝑠 ‘𝑊) 0 ) = 0 ) |
23 | 22 | 3ad2antr1 1185 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊) 0 ) = 0 ) |
24 | 18, 23 | eqtrd 2765 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊)𝑎) = 0 ) |
25 | simpr3 1193 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑏 ∈ { 0 }) | |
26 | elsni 4649 | . . . . . 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 7441 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = ( 0 (+g‘𝑊) 0 )) |
29 | eqid 2725 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
30 | 8, 29, 9 | lmod0vlid 20815 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 0 ∈ (Base‘𝑊)) → ( 0 (+g‘𝑊) 0 ) = 0 ) |
31 | 10, 30 | mpdan 685 | . . . . 5 ⊢ (𝑊 ∈ LMod → ( 0 (+g‘𝑊) 0 ) = 0 ) |
32 | 31 | adantr 479 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ( 0 (+g‘𝑊) 0 ) = 0 ) |
33 | 28, 32 | eqtrd 2765 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
34 | ovex 7456 | . . . 4 ⊢ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ V | |
35 | 34 | elsn 4647 | . . 3 ⊢ (((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ { 0 } ↔ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
36 | 33, 35 | sylibr 233 | . 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 20859 | 1 ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∅c0 4324 {csn 4632 ‘cfv 6553 (class class class)co 7423 Basecbs 17208 +gcplusg 17261 Scalarcsca 17264 ·𝑠 cvsca 17265 0gc0g 17449 LModclmod 20783 LSubSpclss 20855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-plusg 17274 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-minusg 18927 df-cmn 19775 df-abl 19776 df-mgp 20113 df-rng 20131 df-ur 20160 df-ring 20213 df-lmod 20785 df-lss 20856 |
This theorem is referenced by: lspsn0 20932 lsp0 20933 lmhmkerlss 20976 lidl0ALT 21164 lsatcv0 38689 lsatcveq0 38690 lsat0cv 38691 lsatcv0eq 38705 dochsat 41042 mapd0 41324 mapdcnvatN 41325 mapdat 41326 mapdn0 41328 hdmapeq0 41503 |
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