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Mirrors > Home > MPE Home > Th. List > Mathboxes > doch0 | Structured version Visualization version GIF version |
Description: Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.) |
Ref | Expression |
---|---|
doch0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
doch0.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
doch0.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
doch0.v | ⊢ 𝑉 = (Base‘𝑈) |
doch0.z | ⊢ 0 = (0g‘𝑈) |
Ref | Expression |
---|---|
doch0 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | doch0.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2777 | . . . 4 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
3 | doch0.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | doch0.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
5 | 1, 2, 3, 4 | dih0rn 37422 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → { 0 } ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
6 | eqid 2777 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
7 | doch0.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
8 | 6, 1, 2, 7 | dochvalr 37495 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ { 0 } ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘{ 0 }) = (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘{ 0 })))) |
9 | 5, 8 | mpdan 677 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘{ 0 })))) |
10 | eqid 2777 | . . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
11 | 1, 10, 2, 3, 4 | dih0cnv 37421 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (◡((DIsoH‘𝐾)‘𝑊)‘{ 0 }) = (0.‘𝐾)) |
12 | 11 | fveq2d 6450 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘{ 0 })) = ((oc‘𝐾)‘(0.‘𝐾))) |
13 | hlop 35500 | . . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
14 | 13 | adantr 474 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
15 | eqid 2777 | . . . . . . 7 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
16 | 10, 15, 6 | opoc0 35341 | . . . . . 6 ⊢ (𝐾 ∈ OP → ((oc‘𝐾)‘(0.‘𝐾)) = (1.‘𝐾)) |
17 | 14, 16 | syl 17 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((oc‘𝐾)‘(0.‘𝐾)) = (1.‘𝐾)) |
18 | 12, 17 | eqtrd 2813 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘{ 0 })) = (1.‘𝐾)) |
19 | 18 | fveq2d 6450 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘{ 0 }))) = (((DIsoH‘𝐾)‘𝑊)‘(1.‘𝐾))) |
20 | doch0.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
21 | 15, 1, 2, 3, 20 | dih1 37424 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoH‘𝐾)‘𝑊)‘(1.‘𝐾)) = 𝑉) |
22 | 19, 21 | eqtrd 2813 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘{ 0 }))) = 𝑉) |
23 | 9, 22 | eqtrd 2813 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 {csn 4397 ◡ccnv 5354 ran crn 5356 ‘cfv 6135 Basecbs 16255 occoc 16346 0gc0g 16486 0.cp0 17423 1.cp1 17424 OPcops 35310 HLchlt 35488 LHypclh 36122 DVecHcdvh 37216 DIsoHcdih 37366 ocHcoch 37485 |
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 35091 |
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 35314 df-ol 35316 df-oml 35317 df-covers 35404 df-ats 35405 df-atl 35436 df-cvlat 35460 df-hlat 35489 df-llines 35636 df-lplanes 35637 df-lvols 35638 df-lines 35639 df-psubsp 35641 df-pmap 35642 df-padd 35934 df-lhyp 36126 df-laut 36127 df-ldil 36242 df-ltrn 36243 df-trl 36297 df-tendo 36893 df-edring 36895 df-disoa 37167 df-dvech 37217 df-dib 37277 df-dic 37311 df-dih 37367 df-doch 37486 |
This theorem is referenced by: dochoc0 37498 dochoc1 37499 dochsatshp 37589 dochshpsat 37592 dochexmidlem6 37603 dochexmid 37606 lcfl8 37640 lcfl9a 37643 lclkrlem2j 37654 mapd0 37803 hdmaplkr 38051 |
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