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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 2736 | . 2 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
2 | eqid 2736 | . 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
3 | eqid 2736 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
4 | eqid 2736 | . 2 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
5 | eqid 2736 | . 2 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈) | |
6 | dochpol.p | . 2 ⊢ 𝑃 = (LPol‘𝑈) | |
7 | dochpol.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | 7 | fvexi 6853 | . . 3 ⊢ 𝑈 ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 𝑈 ∈ V) |
10 | dochpol.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | eqid 2736 | . . . 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 39786 | . . 3 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶ran ((DIsoH‘𝐾)‘𝑊)) |
15 | 10, 7, 11, 2 | dihsslss 39706 | . . . 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 6683 | . 2 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶(LSubSp‘𝑈)) |
18 | 10, 7, 12, 1, 3 | doch1 39789 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
19 | 13, 18 | syl 17 | . 2 ⊢ (𝜑 → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
20 | 13 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
21 | simpr2 1195 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑦 ⊆ (Base‘𝑈)) | |
22 | simpr3 1196 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑥 ⊆ 𝑦) | |
23 | 10, 7, 1, 12 | dochss 39795 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) |
24 | 20, 21, 22, 23 | syl3anc 1371 | . 2 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) |
25 | 13 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
26 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ (LSAtoms‘𝑈)) | |
27 | 10, 7, 12, 4, 5, 25, 26 | dochsatshp 39881 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘𝑥) ∈ (LSHyp‘𝑈)) |
28 | 10, 7, 11, 4 | dih1dimat 39760 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
29 | 25, 26, 28 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
30 | 10, 11, 12 | dochoc 39797 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) |
31 | 25, 29, 30 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) |
32 | 1, 2, 3, 4, 5, 6, 9, 17, 19, 24, 27, 31 | islpoldN 39914 | 1 ⊢ (𝜑 → ⊥ ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ⊆ wss 3908 𝒫 cpw 4558 {csn 4584 ran crn 5632 ‘cfv 6493 Basecbs 17075 0gc0g 17313 LSubSpclss 20377 LSAtomsclsa 37403 LSHypclsh 37404 HLchlt 37779 LHypclh 38414 DVecHcdvh 39508 DIsoHcdih 39658 ocHcoch 39777 LPolclpoN 39910 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-riotaBAD 37382 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 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-tp 4589 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 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 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-tpos 8153 df-undef 8200 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-sca 17141 df-vsca 17142 df-0g 17315 df-proset 18176 df-poset 18194 df-plt 18211 df-lub 18227 df-glb 18228 df-join 18229 df-meet 18230 df-p0 18306 df-p1 18307 df-lat 18313 df-clat 18380 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-submnd 18594 df-grp 18743 df-minusg 18744 df-sbg 18745 df-subg 18916 df-cntz 19088 df-lsm 19409 df-cmn 19555 df-abl 19556 df-mgp 19888 df-ur 19905 df-ring 19952 df-oppr 20034 df-dvdsr 20055 df-unit 20056 df-invr 20086 df-dvr 20097 df-drng 20172 df-lmod 20309 df-lss 20378 df-lsp 20418 df-lvec 20549 df-lsatoms 37405 df-lshyp 37406 df-oposet 37605 df-ol 37607 df-oml 37608 df-covers 37695 df-ats 37696 df-atl 37727 df-cvlat 37751 df-hlat 37780 df-llines 37928 df-lplanes 37929 df-lvols 37930 df-lines 37931 df-psubsp 37933 df-pmap 37934 df-padd 38226 df-lhyp 38418 df-laut 38419 df-ldil 38534 df-ltrn 38535 df-trl 38589 df-tgrp 39173 df-tendo 39185 df-edring 39187 df-dveca 39433 df-disoa 39459 df-dvech 39509 df-dib 39569 df-dic 39603 df-dih 39659 df-doch 39778 df-djh 39825 df-lpolN 39911 |
This theorem is referenced by: (None) |
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