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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochn0nv | Structured version Visualization version GIF version |
Description: An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.) |
Ref | Expression |
---|---|
dochn0nv.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochn0nv.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochn0nv.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochn0nv.v | ⊢ 𝑉 = (Base‘𝑈) |
dochn0nv.z | ⊢ 0 = (0g‘𝑈) |
dochn0nv.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochn0nv.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
Ref | Expression |
---|---|
dochn0nv | ⊢ (𝜑 → (( ⊥ ‘𝑋) ≠ { 0 } ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochn0nv.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dochn0nv.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) | |
3 | dochn0nv.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | eqid 2777 | . . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
5 | dochn0nv.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | dochn0nv.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
7 | dochn0nv.o | . . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
8 | 3, 4, 5, 6, 7 | dochcl 37502 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
9 | 1, 2, 8 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
10 | 3, 4, 7 | dochoc 37516 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
11 | 1, 9, 10 | syl2anc 579 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
12 | dochn0nv.z | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
13 | 3, 5, 7, 6, 12 | doch1 37508 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑉) = { 0 }) |
14 | 1, 13 | syl 17 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 }) |
15 | 11, 14 | eqeq12d 2792 | . . 3 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑉) ↔ ( ⊥ ‘𝑋) = { 0 })) |
16 | 3, 5, 6, 7 | dochssv 37504 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉) |
17 | 1, 2, 16 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉) |
18 | 3, 4, 5, 6, 7 | dochcl 37502 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
19 | 1, 17, 18 | syl2anc 579 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
20 | 3, 4, 5, 6 | dih1rn 37436 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
21 | 1, 20 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
22 | 3, 4, 7, 1, 19, 21 | doch11 37522 | . . 3 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑉) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑉)) |
23 | 15, 22 | bitr3d 273 | . 2 ⊢ (𝜑 → (( ⊥ ‘𝑋) = { 0 } ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑉)) |
24 | 23 | necon3bid 3012 | 1 ⊢ (𝜑 → (( ⊥ ‘𝑋) ≠ { 0 } ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ⊆ wss 3791 {csn 4397 ran crn 5356 ‘cfv 6135 Basecbs 16255 0gc0g 16486 HLchlt 35499 LHypclh 36133 DVecHcdvh 37227 DIsoHcdih 37377 ocHcoch 37496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35102 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-oposet 35325 df-ol 35327 df-oml 35328 df-covers 35415 df-ats 35416 df-atl 35447 df-cvlat 35471 df-hlat 35500 df-llines 35647 df-lplanes 35648 df-lvols 35649 df-lines 35650 df-psubsp 35652 df-pmap 35653 df-padd 35945 df-lhyp 36137 df-laut 36138 df-ldil 36253 df-ltrn 36254 df-trl 36308 df-tendo 36904 df-edring 36906 df-disoa 37178 df-dvech 37228 df-dib 37288 df-dic 37322 df-dih 37378 df-doch 37497 |
This theorem is referenced by: dochsnnz 37599 dochsatshpb 37601 dochkrsat 37604 dochkrsat2 37605 dochsnkrlem1 37618 |
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