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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvecdim0i | Structured version Visualization version GIF version |
Description: A vector space of dimension zero is reduced to its identity element. (Contributed by Thierry Arnoux, 31-Jul-2023.) |
Ref | Expression |
---|---|
lvecdim0.1 | β’ 0 = (0gβπ) |
Ref | Expression |
---|---|
lvecdim0i | β’ ((π β LVec β§ (dimβπ) = 0) β (Baseβπ) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . . . 7 β’ (LBasisβπ) = (LBasisβπ) | |
2 | 1 | lbsex 20671 | . . . . . 6 β’ (π β LVec β (LBasisβπ) β β ) |
3 | n0 4310 | . . . . . 6 β’ ((LBasisβπ) β β β βπ π β (LBasisβπ)) | |
4 | 2, 3 | sylib 217 | . . . . 5 β’ (π β LVec β βπ π β (LBasisβπ)) |
5 | 4 | adantr 482 | . . . 4 β’ ((π β LVec β§ (dimβπ) = 0) β βπ π β (LBasisβπ)) |
6 | simpr 486 | . . . . . 6 β’ (((π β LVec β§ (dimβπ) = 0) β§ π β (LBasisβπ)) β π β (LBasisβπ)) | |
7 | 1 | dimval 32362 | . . . . . . . 8 β’ ((π β LVec β§ π β (LBasisβπ)) β (dimβπ) = (β―βπ)) |
8 | 7 | adantlr 714 | . . . . . . 7 β’ (((π β LVec β§ (dimβπ) = 0) β§ π β (LBasisβπ)) β (dimβπ) = (β―βπ)) |
9 | simplr 768 | . . . . . . 7 β’ (((π β LVec β§ (dimβπ) = 0) β§ π β (LBasisβπ)) β (dimβπ) = 0) | |
10 | 8, 9 | eqtr3d 2775 | . . . . . 6 β’ (((π β LVec β§ (dimβπ) = 0) β§ π β (LBasisβπ)) β (β―βπ) = 0) |
11 | hasheq0 14272 | . . . . . . 7 β’ (π β (LBasisβπ) β ((β―βπ) = 0 β π = β )) | |
12 | 11 | biimpa 478 | . . . . . 6 β’ ((π β (LBasisβπ) β§ (β―βπ) = 0) β π = β ) |
13 | 6, 10, 12 | syl2anc 585 | . . . . 5 β’ (((π β LVec β§ (dimβπ) = 0) β§ π β (LBasisβπ)) β π = β ) |
14 | 13, 6 | eqeltrrd 2835 | . . . 4 β’ (((π β LVec β§ (dimβπ) = 0) β§ π β (LBasisβπ)) β β β (LBasisβπ)) |
15 | 5, 14 | exlimddv 1939 | . . 3 β’ ((π β LVec β§ (dimβπ) = 0) β β β (LBasisβπ)) |
16 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
17 | eqid 2733 | . . . 4 β’ (LSpanβπ) = (LSpanβπ) | |
18 | 16, 1, 17 | lbssp 20584 | . . 3 β’ (β β (LBasisβπ) β ((LSpanβπ)ββ ) = (Baseβπ)) |
19 | 15, 18 | syl 17 | . 2 β’ ((π β LVec β§ (dimβπ) = 0) β ((LSpanβπ)ββ ) = (Baseβπ)) |
20 | lveclmod 20611 | . . . 4 β’ (π β LVec β π β LMod) | |
21 | 20 | adantr 482 | . . 3 β’ ((π β LVec β§ (dimβπ) = 0) β π β LMod) |
22 | lvecdim0.1 | . . . 4 β’ 0 = (0gβπ) | |
23 | 22, 17 | lsp0 20514 | . . 3 β’ (π β LMod β ((LSpanβπ)ββ ) = { 0 }) |
24 | 21, 23 | syl 17 | . 2 β’ ((π β LVec β§ (dimβπ) = 0) β ((LSpanβπ)ββ ) = { 0 }) |
25 | 19, 24 | eqtr3d 2775 | 1 β’ ((π β LVec β§ (dimβπ) = 0) β (Baseβπ) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 β wne 2940 β c0 4286 {csn 4590 βcfv 6500 0cc0 11059 β―chash 14239 Basecbs 17091 0gc0g 17329 LModclmod 20365 LSpanclspn 20476 LBasisclbs 20579 LVecclvec 20607 dimcldim 32360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-reg 9536 ax-inf2 9585 ax-ac2 10407 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-rpss 7664 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-oi 9454 df-r1 9708 df-rank 9709 df-dju 9845 df-card 9883 df-acn 9886 df-ac 10060 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-xnn0 12494 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-hash 14240 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-tset 17160 df-ple 17161 df-ocomp 17162 df-0g 17331 df-mre 17474 df-mrc 17475 df-mri 17476 df-acs 17477 df-proset 18192 df-drs 18193 df-poset 18210 df-ipo 18425 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-drng 20221 df-lmod 20367 df-lss 20437 df-lsp 20477 df-lbs 20580 df-lvec 20608 df-dim 32361 |
This theorem is referenced by: lvecdim0 32366 |
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