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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochpolN | Structured version Visualization version GIF version |
Description: The subspace orthocomplement for the DVecH vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dochpol.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochpol.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochpol.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochpol.p | ⊢ 𝑃 = (LPol‘𝑈) |
dochpol.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
dochpolN | ⊢ (𝜑 → ⊥ ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2779 | . 2 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
2 | eqid 2779 | . 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
3 | eqid 2779 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
4 | eqid 2779 | . 2 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
5 | eqid 2779 | . 2 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈) | |
6 | dochpol.p | . 2 ⊢ 𝑃 = (LPol‘𝑈) | |
7 | dochpol.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | 7 | fvexi 6513 | . . 3 ⊢ 𝑈 ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 𝑈 ∈ V) |
10 | dochpol.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | eqid 2779 | . . . 4 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
12 | dochpol.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
13 | dochpol.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | 10, 11, 7, 1, 12, 13 | dochfN 37934 | . . 3 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶ran ((DIsoH‘𝐾)‘𝑊)) |
15 | 10, 7, 11, 2 | dihsslss 37854 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran ((DIsoH‘𝐾)‘𝑊) ⊆ (LSubSp‘𝑈)) |
16 | 13, 15 | syl 17 | . . 3 ⊢ (𝜑 → ran ((DIsoH‘𝐾)‘𝑊) ⊆ (LSubSp‘𝑈)) |
17 | 14, 16 | fssd 6358 | . 2 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶(LSubSp‘𝑈)) |
18 | 10, 7, 12, 1, 3 | doch1 37937 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
19 | 13, 18 | syl 17 | . 2 ⊢ (𝜑 → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
20 | 13 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
21 | simpr2 1175 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑦 ⊆ (Base‘𝑈)) | |
22 | simpr3 1176 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑥 ⊆ 𝑦) | |
23 | 10, 7, 1, 12 | dochss 37943 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) |
24 | 20, 21, 22, 23 | syl3anc 1351 | . 2 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) |
25 | 13 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
26 | simpr 477 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ (LSAtoms‘𝑈)) | |
27 | 10, 7, 12, 4, 5, 25, 26 | dochsatshp 38029 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘𝑥) ∈ (LSHyp‘𝑈)) |
28 | 10, 7, 11, 4 | dih1dimat 37908 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
29 | 25, 26, 28 | syl2anc 576 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
30 | 10, 11, 12 | dochoc 37945 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) |
31 | 25, 29, 30 | syl2anc 576 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) |
32 | 1, 2, 3, 4, 5, 6, 9, 17, 19, 24, 27, 31 | islpoldN 38062 | 1 ⊢ (𝜑 → ⊥ ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 Vcvv 3416 ⊆ wss 3830 𝒫 cpw 4422 {csn 4441 ran crn 5408 ‘cfv 6188 Basecbs 16339 0gc0g 16569 LSubSpclss 19425 LSAtomsclsa 35552 LSHypclsh 35553 HLchlt 35928 LHypclh 36562 DVecHcdvh 37656 DIsoHcdih 37806 ocHcoch 37925 LPolclpoN 38058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-riotaBAD 35531 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-tpos 7695 df-undef 7742 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-0g 16571 df-proset 17396 df-poset 17414 df-plt 17426 df-lub 17442 df-glb 17443 df-join 17444 df-meet 17445 df-p0 17507 df-p1 17508 df-lat 17514 df-clat 17576 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-subg 18060 df-cntz 18218 df-lsm 18522 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-dvr 19156 df-drng 19227 df-lmod 19358 df-lss 19426 df-lsp 19466 df-lvec 19597 df-lsatoms 35554 df-lshyp 35555 df-oposet 35754 df-ol 35756 df-oml 35757 df-covers 35844 df-ats 35845 df-atl 35876 df-cvlat 35900 df-hlat 35929 df-llines 36076 df-lplanes 36077 df-lvols 36078 df-lines 36079 df-psubsp 36081 df-pmap 36082 df-padd 36374 df-lhyp 36566 df-laut 36567 df-ldil 36682 df-ltrn 36683 df-trl 36737 df-tgrp 37321 df-tendo 37333 df-edring 37335 df-dveca 37581 df-disoa 37607 df-dvech 37657 df-dib 37717 df-dic 37751 df-dih 37807 df-doch 37926 df-djh 37973 df-lpolN 38059 |
This theorem is referenced by: (None) |
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