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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 2738 | . 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
| 2 | eqidd 2738 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))) | |
| 3 | eqidd 2738 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) = (Base‘𝑊)) | |
| 4 | eqidd 2738 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
| 5 | eqidd 2738 | . 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 2737 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 9 | lss0cl.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
| 10 | 8, 9 | lmod0vcl 20889 | . . 3 ⊢ (𝑊 ∈ LMod → 0 ∈ (Base‘𝑊)) |
| 11 | 10 | snssd 4809 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ⊆ (Base‘𝑊)) |
| 12 | 9 | fvexi 6920 | . . . 4 ⊢ 0 ∈ V |
| 13 | 12 | snnz 4776 | . . 3 ⊢ { 0 } ≠ ∅ |
| 14 | 13 | a1i 11 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ≠ ∅) |
| 15 | simpr2 1196 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑎 ∈ { 0 }) | |
| 16 | elsni 4643 | . . . . . . . 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 7447 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊)𝑎) = (𝑥( ·𝑠 ‘𝑊) 0 )) |
| 19 | eqid 2737 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 20 | eqid 2737 | . . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 21 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 22 | 19, 20, 21, 9 | lmodvs0 20894 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝑥( ·𝑠 ‘𝑊) 0 ) = 0 ) |
| 23 | 22 | 3ad2antr1 1189 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊) 0 ) = 0 ) |
| 24 | 18, 23 | eqtrd 2777 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊)𝑎) = 0 ) |
| 25 | simpr3 1197 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑏 ∈ { 0 }) | |
| 26 | elsni 4643 | . . . . . 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 7449 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = ( 0 (+g‘𝑊) 0 )) |
| 29 | eqid 2737 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 30 | 8, 29, 9 | lmod0vlid 20890 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 0 ∈ (Base‘𝑊)) → ( 0 (+g‘𝑊) 0 ) = 0 ) |
| 31 | 10, 30 | mpdan 687 | . . . . 5 ⊢ (𝑊 ∈ LMod → ( 0 (+g‘𝑊) 0 ) = 0 ) |
| 32 | 31 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ( 0 (+g‘𝑊) 0 ) = 0 ) |
| 33 | 28, 32 | eqtrd 2777 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
| 34 | ovex 7464 | . . . 4 ⊢ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ V | |
| 35 | 34 | elsn 4641 | . . 3 ⊢ (((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ { 0 } ↔ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
| 36 | 33, 35 | sylibr 234 | . 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 20933 | 1 ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 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 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-lmod 20860 df-lss 20930 |
| This theorem is referenced by: lspsn0 21006 lsp0 21007 lmhmkerlss 21050 lidl0ALT 21238 lsatcv0 39032 lsatcveq0 39033 lsat0cv 39034 lsatcv0eq 39048 dochsat 41385 mapd0 41667 mapdcnvatN 41668 mapdat 41669 mapdn0 41671 hdmapeq0 41846 |
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