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Mirrors > Home > MPE Home > Th. List > lspdisjb | Structured version Visualization version GIF version |
Description: A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.) |
Ref | Expression |
---|---|
lspdisjb.v | β’ π = (Baseβπ) |
lspdisjb.o | β’ 0 = (0gβπ) |
lspdisjb.n | β’ π = (LSpanβπ) |
lspdisjb.s | β’ π = (LSubSpβπ) |
lspdisjb.w | β’ (π β π β LVec) |
lspdisjb.u | β’ (π β π β π) |
lspdisjb.x | β’ (π β π β (π β { 0 })) |
Ref | Expression |
---|---|
lspdisjb | β’ (π β (Β¬ π β π β ((πβ{π}) β© π) = { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspdisjb.v | . . 3 β’ π = (Baseβπ) | |
2 | lspdisjb.o | . . 3 β’ 0 = (0gβπ) | |
3 | lspdisjb.n | . . 3 β’ π = (LSpanβπ) | |
4 | lspdisjb.s | . . 3 β’ π = (LSubSpβπ) | |
5 | lspdisjb.w | . . . 4 β’ (π β π β LVec) | |
6 | 5 | adantr 482 | . . 3 β’ ((π β§ Β¬ π β π) β π β LVec) |
7 | lspdisjb.u | . . . 4 β’ (π β π β π) | |
8 | 7 | adantr 482 | . . 3 β’ ((π β§ Β¬ π β π) β π β π) |
9 | lspdisjb.x | . . . . 5 β’ (π β π β (π β { 0 })) | |
10 | 9 | eldifad 3926 | . . . 4 β’ (π β π β π) |
11 | 10 | adantr 482 | . . 3 β’ ((π β§ Β¬ π β π) β π β π) |
12 | simpr 486 | . . 3 β’ ((π β§ Β¬ π β π) β Β¬ π β π) | |
13 | 1, 2, 3, 4, 6, 8, 11, 12 | lspdisj 20631 | . 2 β’ ((π β§ Β¬ π β π) β ((πβ{π}) β© π) = { 0 }) |
14 | eldifsni 4754 | . . . . 5 β’ (π β (π β { 0 }) β π β 0 ) | |
15 | 9, 14 | syl 17 | . . . 4 β’ (π β π β 0 ) |
16 | 15 | adantr 482 | . . 3 β’ ((π β§ ((πβ{π}) β© π) = { 0 }) β π β 0 ) |
17 | lveclmod 20611 | . . . . . . 7 β’ (π β LVec β π β LMod) | |
18 | 5, 17 | syl 17 | . . . . . 6 β’ (π β π β LMod) |
19 | 1, 3 | lspsnid 20498 | . . . . . 6 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
20 | 18, 10, 19 | syl2anc 585 | . . . . 5 β’ (π β π β (πβ{π})) |
21 | elin 3930 | . . . . . . 7 β’ (π β ((πβ{π}) β© π) β (π β (πβ{π}) β§ π β π)) | |
22 | eleq2 2823 | . . . . . . . 8 β’ (((πβ{π}) β© π) = { 0 } β (π β ((πβ{π}) β© π) β π β { 0 })) | |
23 | elsni 4607 | . . . . . . . 8 β’ (π β { 0 } β π = 0 ) | |
24 | 22, 23 | syl6bi 253 | . . . . . . 7 β’ (((πβ{π}) β© π) = { 0 } β (π β ((πβ{π}) β© π) β π = 0 )) |
25 | 21, 24 | biimtrrid 242 | . . . . . 6 β’ (((πβ{π}) β© π) = { 0 } β ((π β (πβ{π}) β§ π β π) β π = 0 )) |
26 | 25 | expd 417 | . . . . 5 β’ (((πβ{π}) β© π) = { 0 } β (π β (πβ{π}) β (π β π β π = 0 ))) |
27 | 20, 26 | mpan9 508 | . . . 4 β’ ((π β§ ((πβ{π}) β© π) = { 0 }) β (π β π β π = 0 )) |
28 | 27 | necon3ad 2953 | . . 3 β’ ((π β§ ((πβ{π}) β© π) = { 0 }) β (π β 0 β Β¬ π β π)) |
29 | 16, 28 | mpd 15 | . 2 β’ ((π β§ ((πβ{π}) β© π) = { 0 }) β Β¬ π β π) |
30 | 13, 29 | impbida 800 | 1 β’ (π β (Β¬ π β π β ((πβ{π}) β© π) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 β cdif 3911 β© cin 3913 {csn 4590 βcfv 6500 Basecbs 17091 0gc0g 17329 LModclmod 20365 LSubSpclss 20436 LSpanclspn 20476 LVecclvec 20607 |
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-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-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-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-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 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-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 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-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-sbg 18761 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-lvec 20608 |
This theorem is referenced by: mapdh6b0N 40249 hdmap1l6b0N 40323 |
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