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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 2799 | . 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
2 | eqidd 2799 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))) | |
3 | eqidd 2799 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) = (Base‘𝑊)) | |
4 | eqidd 2799 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
5 | eqidd 2799 | . 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 2798 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
9 | lss0cl.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
10 | 8, 9 | lmod0vcl 19656 | . . 3 ⊢ (𝑊 ∈ LMod → 0 ∈ (Base‘𝑊)) |
11 | 10 | snssd 4702 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ⊆ (Base‘𝑊)) |
12 | 9 | fvexi 6659 | . . . 4 ⊢ 0 ∈ V |
13 | 12 | snnz 4672 | . . 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 4542 | . . . . . . . 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 7151 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊)𝑎) = (𝑥( ·𝑠 ‘𝑊) 0 )) |
19 | eqid 2798 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
20 | eqid 2798 | . . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
21 | eqid 2798 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
22 | 19, 20, 21, 9 | lmodvs0 19661 | . . . . . . 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 2833 | . . . . 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 4542 | . . . . . 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 7153 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = ( 0 (+g‘𝑊) 0 )) |
29 | eqid 2798 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
30 | 8, 29, 9 | lmod0vlid 19657 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 0 ∈ (Base‘𝑊)) → ( 0 (+g‘𝑊) 0 ) = 0 ) |
31 | 10, 30 | mpdan 686 | . . . . 5 ⊢ (𝑊 ∈ LMod → ( 0 (+g‘𝑊) 0 ) = 0 ) |
32 | 31 | adantr 484 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ( 0 (+g‘𝑊) 0 ) = 0 ) |
33 | 28, 32 | eqtrd 2833 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
34 | ovex 7168 | . . . 4 ⊢ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ V | |
35 | 34 | elsn 4540 | . . 3 ⊢ (((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ { 0 } ↔ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
36 | 33, 35 | sylibr 237 | . 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 19700 | 1 ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 LModclmod 19627 LSubSpclss 19696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-mgp 19233 df-ring 19292 df-lmod 19629 df-lss 19697 |
This theorem is referenced by: lspsn0 19773 lsp0 19774 lmhmkerlss 19816 lidl0 19985 lsatcv0 36327 lsatcveq0 36328 lsat0cv 36329 lsatcv0eq 36343 dochsat 38679 mapd0 38961 mapdcnvatN 38962 mapdat 38963 mapdn0 38965 hdmapeq0 39140 |
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