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Mathbox for Norm Megill |
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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 36117 analog.) TODO: This changes 𝑈𝐶𝑉 in l1cvpat 35128 and l1cvat 35129 to 𝑈 ∈ 𝐻, which in turn change 𝑈 ∈ 𝐻 in islshpcv 35127 to 𝑈𝐶𝑉, 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 | ⊢ 𝑆 = (LSubSp‘𝑊) |
lshpat.p | ⊢ ⊕ = (LSSum‘𝑊) |
ishpat.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lshpat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lshpat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lshpat.l | ⊢ (𝜑 → 𝑈 ∈ 𝐻) |
lshpat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lshpat.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
lshpat.n | ⊢ (𝜑 → 𝑄 ≠ 𝑅) |
lshpat.m | ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) |
Ref | Expression |
---|---|
lshpat | ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . 2 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lshpat.s | . 2 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lshpat.p | . 2 ⊢ ⊕ = (LSSum‘𝑊) | |
4 | lshpat.a | . 2 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
5 | eqid 2825 | . 2 ⊢ ( ⋖L ‘𝑊) = ( ⋖L ‘𝑊) | |
6 | lshpat.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | lshpat.l | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
8 | ishpat.h | . . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊) | |
9 | 1, 2, 8, 5, 6 | islshpcv 35127 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈( ⋖L ‘𝑊)(Base‘𝑊)))) |
10 | 7, 9 | mpbid 224 | . . 3 ⊢ (𝜑 → (𝑈 ∈ 𝑆 ∧ 𝑈( ⋖L ‘𝑊)(Base‘𝑊))) |
11 | 10 | simpld 490 | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
12 | lshpat.q | . 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
13 | lshpat.r | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
14 | lshpat.n | . 2 ⊢ (𝜑 → 𝑄 ≠ 𝑅) | |
15 | 10 | simprd 491 | . 2 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(Base‘𝑊)) |
16 | lshpat.m | . 2 ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) | |
17 | 1, 2, 3, 4, 5, 6, 11, 12, 13, 14, 15, 16 | l1cvat 35129 | 1 ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∩ cin 3797 ⊆ wss 3798 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 LSSumclsm 18407 LSubSpclss 19295 LVecclvec 19468 LSAtomsclsa 35048 LSHypclsh 35049 ⋖L clcv 35092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-0g 16462 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cntz 18107 df-oppg 18133 df-lsm 18409 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 df-lsatoms 35050 df-lshyp 35051 df-lcv 35093 |
This theorem is referenced by: lclkrlem2a 37581 lcfrlem20 37636 |
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