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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 2725 | . 2 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
2 | eqid 2725 | . 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
3 | eqid 2725 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
4 | eqid 2725 | . 2 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
5 | eqid 2725 | . 2 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈) | |
6 | dochpol.p | . 2 ⊢ 𝑃 = (LPol‘𝑈) | |
7 | dochpol.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | 7 | fvexi 6905 | . . 3 ⊢ 𝑈 ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 𝑈 ∈ V) |
10 | dochpol.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | eqid 2725 | . . . 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 40884 | . . 3 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶ran ((DIsoH‘𝐾)‘𝑊)) |
15 | 10, 7, 11, 2 | dihsslss 40804 | . . . 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 6734 | . 2 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶(LSubSp‘𝑈)) |
18 | 10, 7, 12, 1, 3 | doch1 40887 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
19 | 13, 18 | syl 17 | . 2 ⊢ (𝜑 → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
20 | 13 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
21 | simpr2 1192 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑦 ⊆ (Base‘𝑈)) | |
22 | simpr3 1193 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑥 ⊆ 𝑦) | |
23 | 10, 7, 1, 12 | dochss 40893 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) |
24 | 20, 21, 22, 23 | syl3anc 1368 | . 2 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) |
25 | 13 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
26 | simpr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ (LSAtoms‘𝑈)) | |
27 | 10, 7, 12, 4, 5, 25, 26 | dochsatshp 40979 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘𝑥) ∈ (LSHyp‘𝑈)) |
28 | 10, 7, 11, 4 | dih1dimat 40858 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
29 | 25, 26, 28 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
30 | 10, 11, 12 | dochoc 40895 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) |
31 | 25, 29, 30 | syl2anc 582 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) |
32 | 1, 2, 3, 4, 5, 6, 9, 17, 19, 24, 27, 31 | islpoldN 41012 | 1 ⊢ (𝜑 → ⊥ ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3940 𝒫 cpw 4598 {csn 4624 ran crn 5673 ‘cfv 6542 Basecbs 17177 0gc0g 17418 LSubSpclss 20817 LSAtomsclsa 38501 LSHypclsh 38502 HLchlt 38877 LHypclh 39512 DVecHcdvh 40606 DIsoHcdih 40756 ocHcoch 40875 LPolclpoN 41008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-riotaBAD 38480 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-undef 8275 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-0g 17420 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-cntz 19270 df-lsm 19593 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-drng 20628 df-lmod 20747 df-lss 20818 df-lsp 20858 df-lvec 20990 df-lsatoms 38503 df-lshyp 38504 df-oposet 38703 df-ol 38705 df-oml 38706 df-covers 38793 df-ats 38794 df-atl 38825 df-cvlat 38849 df-hlat 38878 df-llines 39026 df-lplanes 39027 df-lvols 39028 df-lines 39029 df-psubsp 39031 df-pmap 39032 df-padd 39324 df-lhyp 39516 df-laut 39517 df-ldil 39632 df-ltrn 39633 df-trl 39687 df-tgrp 40271 df-tendo 40283 df-edring 40285 df-dveca 40531 df-disoa 40557 df-dvech 40607 df-dib 40667 df-dic 40701 df-dih 40757 df-doch 40876 df-djh 40923 df-lpolN 41009 |
This theorem is referenced by: (None) |
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