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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 479 | . . 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 479 | . . 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 3961 | . . . 4 β’ (π β π β π) |
| 11 | 10 | adantr 479 | . . 3 β’ ((π β§ Β¬ π β π) β π β π) |
| 12 | simpr 483 | . . 3 β’ ((π β§ Β¬ π β π) β Β¬ π β π) | |
| 13 | 1, 2, 3, 4, 6, 8, 11, 12 | lspdisj 21020 | . 2 β’ ((π β§ Β¬ π β π) β ((πβ{π}) β© π) = { 0 }) |
| 14 | eldifsni 4798 | . . . . 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 479 | . . 3 β’ ((π β§ ((πβ{π}) β© π) = { 0 }) β π β 0 ) |
| 17 | lveclmod 20998 | . . . . . . 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 20884 | . . . . . 6 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
| 20 | 18, 10, 19 | syl2anc 582 | . . . . 5 β’ (π β π β (πβ{π})) |
| 21 | elin 3965 | . . . . . . 7 β’ (π β ((πβ{π}) β© π) β (π β (πβ{π}) β§ π β π)) | |
| 22 | eleq2 2818 | . . . . . . . 8 β’ (((πβ{π}) β© π) = { 0 } β (π β ((πβ{π}) β© π) β π β { 0 })) | |
| 23 | elsni 4649 | . . . . . . . 8 β’ (π β { 0 } β π = 0 ) | |
| 24 | 22, 23 | biimtrdi 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 414 | . . . . 5 β’ (((πβ{π}) β© π) = { 0 } β (π β (πβ{π}) β (π β π β π = 0 ))) |
| 27 | 20, 26 | mpan9 505 | . . . 4 β’ ((π β§ ((πβ{π}) β© π) = { 0 }) β (π β π β π = 0 )) |
| 28 | 27 | necon3ad 2950 | . . 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 394 = wceq 1533 β wcel 2098 β wne 2937 β cdif 3946 β© cin 3948 {csn 4632 βcfv 6553 Basecbs 17187 0gc0g 17428 LModclmod 20750 LSubSpclss 20822 LSpanclspn 20862 LVecclvec 20994 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 |
| This theorem is referenced by: mapdh6b0N 41241 hdmap1l6b0N 41315 |
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