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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 38392 analog.) TODO: This changes ππΆπ in l1cvpat 37402 and l1cvat 37403 to π β π», which in turn change π β π» in islshpcv 37401 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 2738 | . 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 2738 | . 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 37401 | . . . 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 496 | . 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 497 | . 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 37403 | 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 397 = wceq 1542 β wcel 2107 β wne 2942 β© cin 3908 β wss 3909 class class class wbr 5104 βcfv 6492 (class class class)co 7350 Basecbs 17018 LSSumclsm 19345 LSubSpclss 20315 LVecclvec 20486 LSAtomsclsa 37322 LSHypclsh 37323 βL clcv 37366 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-tpos 8125 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-0g 17258 df-mre 17401 df-mrc 17402 df-acs 17404 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-submnd 18537 df-grp 18686 df-minusg 18687 df-sbg 18688 df-subg 18858 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 37324 df-lshyp 37325 df-lcv 37367 |
This theorem is referenced by: lclkrlem2a 39856 lcfrlem20 39911 |
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