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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpat | Structured version Visualization version GIF version |
Description: Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 38401 analog.) TODO: This changes ππΆπ in l1cvpat 37411 and l1cvat 37412 to π β π», which in turn change π β π» in islshpcv 37410 to ππΆπ, with a couple of conversions of span to atom. Seems convoluted. Would a direct proof be better? (Contributed by NM, 11-Jan-2015.) |
Ref | Expression |
---|---|
lshpat.s | β’ π = (LSubSpβπ) |
lshpat.p | β’ β = (LSSumβπ) |
ishpat.h | β’ π» = (LSHypβπ) |
lshpat.a | β’ π΄ = (LSAtomsβπ) |
lshpat.w | β’ (π β π β LVec) |
lshpat.l | β’ (π β π β π») |
lshpat.q | β’ (π β π β π΄) |
lshpat.r | β’ (π β π β π΄) |
lshpat.n | β’ (π β π β π ) |
lshpat.m | β’ (π β Β¬ π β π) |
Ref | Expression |
---|---|
lshpat | β’ (π β ((π β π ) β© π) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . 2 β’ (Baseβπ) = (Baseβπ) | |
2 | lshpat.s | . 2 β’ π = (LSubSpβπ) | |
3 | lshpat.p | . 2 β’ β = (LSSumβπ) | |
4 | lshpat.a | . 2 β’ π΄ = (LSAtomsβπ) | |
5 | eqid 2737 | . 2 β’ ( βL βπ) = ( βL βπ) | |
6 | lshpat.w | . 2 β’ (π β π β LVec) | |
7 | lshpat.l | . . . 4 β’ (π β π β π») | |
8 | ishpat.h | . . . . 5 β’ π» = (LSHypβπ) | |
9 | 1, 2, 8, 5, 6 | islshpcv 37410 | . . . 4 β’ (π β (π β π» β (π β π β§ π( βL βπ)(Baseβπ)))) |
10 | 7, 9 | mpbid 231 | . . 3 β’ (π β (π β π β§ π( βL βπ)(Baseβπ))) |
11 | 10 | simpld 495 | . 2 β’ (π β π β π) |
12 | lshpat.q | . 2 β’ (π β π β π΄) | |
13 | lshpat.r | . 2 β’ (π β π β π΄) | |
14 | lshpat.n | . 2 β’ (π β π β π ) | |
15 | 10 | simprd 496 | . 2 β’ (π β π( βL βπ)(Baseβπ)) |
16 | lshpat.m | . 2 β’ (π β Β¬ π β π) | |
17 | 1, 2, 3, 4, 5, 6, 11, 12, 13, 14, 15, 16 | l1cvat 37412 | 1 β’ (π β ((π β π ) β© π) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2941 β© cin 3907 β wss 3908 class class class wbr 5103 βcfv 6491 (class class class)co 7349 Basecbs 17017 LSSumclsm 19345 LSubSpclss 20315 LVecclvec 20486 LSAtomsclsa 37331 LSHypclsh 37332 βL clcv 37375 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-tpos 8124 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-3 12150 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-ress 17047 df-plusg 17080 df-mulr 17081 df-0g 17257 df-mre 17400 df-mrc 17401 df-acs 17403 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-submnd 18536 df-grp 18685 df-minusg 18686 df-sbg 18687 df-subg 18857 df-cntz 19029 df-oppg 19056 df-lsm 19347 df-cmn 19493 df-abl 19494 df-mgp 19826 df-ur 19843 df-ring 19890 df-oppr 19972 df-dvdsr 19993 df-unit 19994 df-invr 20024 df-drng 20110 df-lmod 20247 df-lss 20316 df-lsp 20356 df-lvec 20487 df-lsatoms 37333 df-lshyp 37334 df-lcv 37376 |
This theorem is referenced by: lclkrlem2a 39865 lcfrlem20 39920 |
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