![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lvecdim0 | Structured version Visualization version GIF version |
Description: A vector space of dimension zero is reduced to its identity element, biconditional version. (Contributed by Thierry Arnoux, 31-Jul-2023.) |
Ref | Expression |
---|---|
lvecdim0.1 | ⊢ 0 = (0g‘𝑉) |
Ref | Expression |
---|---|
lvecdim0 | ⊢ (𝑉 ∈ LVec → ((dim‘𝑉) = 0 ↔ (Base‘𝑉) = { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecdim0.1 | . . 3 ⊢ 0 = (0g‘𝑉) | |
2 | 1 | lvecdim0i 31092 | . 2 ⊢ ((𝑉 ∈ LVec ∧ (dim‘𝑉) = 0) → (Base‘𝑉) = { 0 }) |
3 | simpl 486 | . . . 4 ⊢ ((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) → 𝑉 ∈ LVec) | |
4 | eqid 2798 | . . . . . . . 8 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉) | |
5 | 4 | lbsex 19930 | . . . . . . 7 ⊢ (𝑉 ∈ LVec → (LBasis‘𝑉) ≠ ∅) |
6 | n0 4260 | . . . . . . 7 ⊢ ((LBasis‘𝑉) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (LBasis‘𝑉)) | |
7 | 5, 6 | sylib 221 | . . . . . 6 ⊢ (𝑉 ∈ LVec → ∃𝑏 𝑏 ∈ (LBasis‘𝑉)) |
8 | 3, 7 | syl 17 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) → ∃𝑏 𝑏 ∈ (LBasis‘𝑉)) |
9 | 1 | fvexi 6659 | . . . . . . . . . 10 ⊢ 0 ∈ V |
10 | 9 | snid 4561 | . . . . . . . . 9 ⊢ 0 ∈ { 0 } |
11 | simpr 488 | . . . . . . . . 9 ⊢ ((((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑏 = { 0 }) → 𝑏 = { 0 }) | |
12 | 10, 11 | eleqtrrid 2897 | . . . . . . . 8 ⊢ ((((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑏 = { 0 }) → 0 ∈ 𝑏) |
13 | simplll 774 | . . . . . . . . 9 ⊢ ((((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑏 = { 0 }) → 𝑉 ∈ LVec) | |
14 | 4 | lbslinds 20522 | . . . . . . . . . 10 ⊢ (LBasis‘𝑉) ⊆ (LIndS‘𝑉) |
15 | simplr 768 | . . . . . . . . . 10 ⊢ ((((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑏 = { 0 }) → 𝑏 ∈ (LBasis‘𝑉)) | |
16 | 14, 15 | sseldi 3913 | . . . . . . . . 9 ⊢ ((((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑏 = { 0 }) → 𝑏 ∈ (LIndS‘𝑉)) |
17 | 1 | 0nellinds 30986 | . . . . . . . . 9 ⊢ ((𝑉 ∈ LVec ∧ 𝑏 ∈ (LIndS‘𝑉)) → ¬ 0 ∈ 𝑏) |
18 | 13, 16, 17 | syl2anc 587 | . . . . . . . 8 ⊢ ((((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) ∧ 𝑏 = { 0 }) → ¬ 0 ∈ 𝑏) |
19 | 12, 18 | pm2.65da 816 | . . . . . . 7 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → ¬ 𝑏 = { 0 }) |
20 | simpr 488 | . . . . . . . . . . . 12 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → 𝑏 ∈ (LBasis‘𝑉)) | |
21 | eqid 2798 | . . . . . . . . . . . . 13 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
22 | 21, 4 | lbsss 19842 | . . . . . . . . . . . 12 ⊢ (𝑏 ∈ (LBasis‘𝑉) → 𝑏 ⊆ (Base‘𝑉)) |
23 | 20, 22 | syl 17 | . . . . . . . . . . 11 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → 𝑏 ⊆ (Base‘𝑉)) |
24 | simplr 768 | . . . . . . . . . . 11 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → (Base‘𝑉) = { 0 }) | |
25 | 23, 24 | sseqtrd 3955 | . . . . . . . . . 10 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → 𝑏 ⊆ { 0 }) |
26 | sssn 4719 | . . . . . . . . . 10 ⊢ (𝑏 ⊆ { 0 } ↔ (𝑏 = ∅ ∨ 𝑏 = { 0 })) | |
27 | 25, 26 | sylib 221 | . . . . . . . . 9 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → (𝑏 = ∅ ∨ 𝑏 = { 0 })) |
28 | 27 | orcomd 868 | . . . . . . . 8 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → (𝑏 = { 0 } ∨ 𝑏 = ∅)) |
29 | 28 | ord 861 | . . . . . . 7 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → (¬ 𝑏 = { 0 } → 𝑏 = ∅)) |
30 | 19, 29 | mpd 15 | . . . . . 6 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → 𝑏 = ∅) |
31 | 30, 20 | eqeltrrd 2891 | . . . . 5 ⊢ (((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) ∧ 𝑏 ∈ (LBasis‘𝑉)) → ∅ ∈ (LBasis‘𝑉)) |
32 | 8, 31 | exlimddv 1936 | . . . 4 ⊢ ((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) → ∅ ∈ (LBasis‘𝑉)) |
33 | 4 | dimval 31089 | . . . 4 ⊢ ((𝑉 ∈ LVec ∧ ∅ ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘∅)) |
34 | 3, 32, 33 | syl2anc 587 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) → (dim‘𝑉) = (♯‘∅)) |
35 | hash0 13724 | . . 3 ⊢ (♯‘∅) = 0 | |
36 | 34, 35 | eqtrdi 2849 | . 2 ⊢ ((𝑉 ∈ LVec ∧ (Base‘𝑉) = { 0 }) → (dim‘𝑉) = 0) |
37 | 2, 36 | impbida 800 | 1 ⊢ (𝑉 ∈ LVec → ((dim‘𝑉) = 0 ↔ (Base‘𝑉) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 {csn 4525 ‘cfv 6324 0cc0 10526 ♯chash 13686 Basecbs 16475 0gc0g 16705 LBasisclbs 19839 LVecclvec 19867 LIndSclinds 20494 dimcldim 31087 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 ax-ac2 9874 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-int 4839 df-iun 4883 df-iin 4884 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-se 5479 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-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-rpss 7429 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-r1 9177 df-rank 9178 df-dju 9314 df-card 9352 df-acn 9355 df-ac 9527 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-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-tset 16576 df-ple 16577 df-ocomp 16578 df-0g 16707 df-mre 16849 df-mrc 16850 df-mri 16851 df-acs 16852 df-proset 17530 df-drs 17531 df-poset 17548 df-ipo 17754 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lbs 19840 df-lvec 19868 df-lindf 20495 df-linds 20496 df-dim 31088 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |