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 2823 | . 2 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
2 | eqid 2823 | . 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
3 | eqid 2823 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
4 | eqid 2823 | . 2 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
5 | eqid 2823 | . 2 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈) | |
6 | dochpol.p | . 2 ⊢ 𝑃 = (LPol‘𝑈) | |
7 | dochpol.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | 7 | fvexi 6686 | . . 3 ⊢ 𝑈 ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 𝑈 ∈ V) |
10 | dochpol.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | eqid 2823 | . . . 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 38494 | . . 3 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶ran ((DIsoH‘𝐾)‘𝑊)) |
15 | 10, 7, 11, 2 | dihsslss 38414 | . . . 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 6530 | . 2 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶(LSubSp‘𝑈)) |
18 | 10, 7, 12, 1, 3 | doch1 38497 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
19 | 13, 18 | syl 17 | . 2 ⊢ (𝜑 → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
20 | 13 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
21 | simpr2 1191 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑦 ⊆ (Base‘𝑈)) | |
22 | simpr3 1192 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑥 ⊆ 𝑦) | |
23 | 10, 7, 1, 12 | dochss 38503 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) |
24 | 20, 21, 22, 23 | syl3anc 1367 | . 2 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) |
25 | 13 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
26 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ (LSAtoms‘𝑈)) | |
27 | 10, 7, 12, 4, 5, 25, 26 | dochsatshp 38589 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘𝑥) ∈ (LSHyp‘𝑈)) |
28 | 10, 7, 11, 4 | dih1dimat 38468 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
29 | 25, 26, 28 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
30 | 10, 11, 12 | dochoc 38505 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) |
31 | 25, 29, 30 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) |
32 | 1, 2, 3, 4, 5, 6, 9, 17, 19, 24, 27, 31 | islpoldN 38622 | 1 ⊢ (𝜑 → ⊥ ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 {csn 4569 ran crn 5558 ‘cfv 6357 Basecbs 16485 0gc0g 16715 LSubSpclss 19705 LSAtomsclsa 36112 LSHypclsh 36113 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 DIsoHcdih 38366 ocHcoch 38485 LPolclpoN 38618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 df-lpolN 38619 |
This theorem is referenced by: (None) |
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