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Mirrors > Home > MPE Home > Th. List > lspindp2 | Structured version Visualization version GIF version |
Description: Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.) |
Ref | Expression |
---|---|
lspindp1.v | β’ π = (Baseβπ) |
lspindp1.o | β’ 0 = (0gβπ) |
lspindp1.n | β’ π = (LSpanβπ) |
lspindp1.w | β’ (π β π β LVec) |
lspindp2.x | β’ (π β π β π) |
lspindp2.y | β’ (π β π β (π β { 0 })) |
lspindp2.z | β’ (π β π β π) |
lspindp2.q | β’ (π β (πβ{π}) β (πβ{π})) |
lspindp2.e | β’ (π β Β¬ π β (πβ{π, π})) |
Ref | Expression |
---|---|
lspindp2 | β’ (π β ((πβ{π}) β (πβ{π}) β§ Β¬ π β (πβ{π, π}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspindp1.v | . 2 β’ π = (Baseβπ) | |
2 | lspindp1.o | . 2 β’ 0 = (0gβπ) | |
3 | lspindp1.n | . 2 β’ π = (LSpanβπ) | |
4 | lspindp1.w | . 2 β’ (π β π β LVec) | |
5 | lspindp2.y | . 2 β’ (π β π β (π β { 0 })) | |
6 | lspindp2.x | . 2 β’ (π β π β π) | |
7 | lspindp2.z | . 2 β’ (π β π β π) | |
8 | lspindp2.q | . . 3 β’ (π β (πβ{π}) β (πβ{π})) | |
9 | 8 | necomd 2993 | . 2 β’ (π β (πβ{π}) β (πβ{π})) |
10 | lspindp2.e | . . 3 β’ (π β Β¬ π β (πβ{π, π})) | |
11 | prcom 4741 | . . . . 5 β’ {π, π} = {π, π} | |
12 | 11 | fveq2i 6905 | . . . 4 β’ (πβ{π, π}) = (πβ{π, π}) |
13 | 12 | eleq2i 2821 | . . 3 β’ (π β (πβ{π, π}) β π β (πβ{π, π})) |
14 | 10, 13 | sylnib 327 | . 2 β’ (π β Β¬ π β (πβ{π, π})) |
15 | 1, 2, 3, 4, 5, 6, 7, 9, 14 | lspindp1 21035 | 1 β’ (π β ((πβ{π}) β (πβ{π}) β§ Β¬ π β (πβ{π, π}))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 β cdif 3946 {csn 4632 {cpr 4634 βcfv 6553 Basecbs 17189 0gc0g 17430 LSpanclspn 20869 LVecclvec 21001 |
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 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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 7879 df-1st 8001 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-cntz 19282 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-drng 20640 df-lmod 20759 df-lss 20830 df-lsp 20870 df-lvec 21002 |
This theorem is referenced by: mapdheq4lem 41244 mapdheq4 41245 mapdh6lem1N 41246 mapdh6lem2N 41247 mapdh6aN 41248 hdmap1l6lem1 41320 hdmap1l6lem2 41321 hdmap1l6a 41322 |
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