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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 481 | . . 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 481 | . . 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 3960 | . . . 4 β’ (π β π β π) |
| 11 | 10 | adantr 481 | . . 3 β’ ((π β§ Β¬ π β π) β π β π) |
| 12 | simpr 485 | . . 3 β’ ((π β§ Β¬ π β π) β Β¬ π β π) | |
| 13 | 1, 2, 3, 4, 6, 8, 11, 12 | lspdisj 20737 | . 2 β’ ((π β§ Β¬ π β π) β ((πβ{π}) β© π) = { 0 }) |
| 14 | eldifsni 4793 | . . . . 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 481 | . . 3 β’ ((π β§ ((πβ{π}) β© π) = { 0 }) β π β 0 ) |
| 17 | lveclmod 20716 | . . . . . . 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 20603 | . . . . . 6 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
| 20 | 18, 10, 19 | syl2anc 584 | . . . . 5 β’ (π β π β (πβ{π})) |
| 21 | elin 3964 | . . . . . . 7 β’ (π β ((πβ{π}) β© π) β (π β (πβ{π}) β§ π β π)) | |
| 22 | eleq2 2822 | . . . . . . . 8 β’ (((πβ{π}) β© π) = { 0 } β (π β ((πβ{π}) β© π) β π β { 0 })) | |
| 23 | elsni 4645 | . . . . . . . 8 β’ (π β { 0 } β π = 0 ) | |
| 24 | 22, 23 | syl6bi 252 | . . . . . . 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 416 | . . . . 5 β’ (((πβ{π}) β© π) = { 0 } β (π β (πβ{π}) β (π β π β π = 0 ))) |
| 27 | 20, 26 | mpan9 507 | . . . 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 799 | 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 396 = wceq 1541 β wcel 2106 β wne 2940 β cdif 3945 β© cin 3947 {csn 4628 βcfv 6543 Basecbs 17143 0gc0g 17384 LModclmod 20470 LSubSpclss 20541 LSpanclspn 20581 LVecclvec 20712 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mgp 19987 df-ur 20004 df-ring 20057 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-drng 20358 df-lmod 20472 df-lss 20542 df-lsp 20582 df-lvec 20713 |
| This theorem is referenced by: mapdh6b0N 40602 hdmap1l6b0N 40676 |
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