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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 2822 | . 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
2 | eqidd 2822 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))) | |
3 | eqidd 2822 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) = (Base‘𝑊)) | |
4 | eqidd 2822 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
5 | eqidd 2822 | . 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 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
9 | lss0cl.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
10 | 8, 9 | lmod0vcl 19663 | . . 3 ⊢ (𝑊 ∈ LMod → 0 ∈ (Base‘𝑊)) |
11 | 10 | snssd 4742 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ⊆ (Base‘𝑊)) |
12 | 9 | fvexi 6684 | . . . 4 ⊢ 0 ∈ V |
13 | 12 | snnz 4711 | . . 3 ⊢ { 0 } ≠ ∅ |
14 | 13 | a1i 11 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ≠ ∅) |
15 | simpr2 1191 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑎 ∈ { 0 }) | |
16 | elsni 4584 | . . . . . . . 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 7172 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊)𝑎) = (𝑥( ·𝑠 ‘𝑊) 0 )) |
19 | eqid 2821 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
20 | eqid 2821 | . . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
21 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
22 | 19, 20, 21, 9 | lmodvs0 19668 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝑥( ·𝑠 ‘𝑊) 0 ) = 0 ) |
23 | 22 | 3ad2antr1 1184 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊) 0 ) = 0 ) |
24 | 18, 23 | eqtrd 2856 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → (𝑥( ·𝑠 ‘𝑊)𝑎) = 0 ) |
25 | simpr3 1192 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → 𝑏 ∈ { 0 }) | |
26 | elsni 4584 | . . . . . 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 7174 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = ( 0 (+g‘𝑊) 0 )) |
29 | eqid 2821 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
30 | 8, 29, 9 | lmod0vlid 19664 | . . . . . 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 483 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ( 0 (+g‘𝑊) 0 ) = 0 ) |
33 | 28, 32 | eqtrd 2856 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ { 0 } ∧ 𝑏 ∈ { 0 })) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
34 | ovex 7189 | . . . 4 ⊢ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ V | |
35 | 34 | elsn 4582 | . . 3 ⊢ (((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ { 0 } ↔ ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) = 0 ) |
36 | 33, 35 | sylibr 236 | . 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 19707 | 1 ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 {csn 4567 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 LModclmod 19634 LSubSpclss 19703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-mgp 19240 df-ring 19299 df-lmod 19636 df-lss 19704 |
This theorem is referenced by: lspsn0 19780 lsp0 19781 lmhmkerlss 19823 lidl0 19992 lsatcv0 36182 lsatcveq0 36183 lsat0cv 36184 lsatcv0eq 36198 dochsat 38534 mapd0 38816 mapdcnvatN 38817 mapdat 38818 mapdn0 38820 hdmapeq0 38995 |
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