Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > doch1 | Structured version Visualization version GIF version |
Description: Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.) |
Ref | Expression |
---|---|
doch1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
doch1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
doch1.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
doch1.v | ⊢ 𝑉 = (Base‘𝑈) |
doch1.z | ⊢ 0 = (0g‘𝑈) |
Ref | Expression |
---|---|
doch1 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑉) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | doch1.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2739 | . . . 4 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
3 | doch1.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | doch1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
5 | 1, 2, 3, 4 | dih1rn 38957 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
6 | eqid 2739 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
7 | doch1.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
8 | 6, 1, 2, 7 | dochvalr 39027 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘𝑉) = (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑉)))) |
9 | 5, 8 | mpdan 687 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑉) = (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑉)))) |
10 | eqid 2739 | . . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
11 | 1, 10, 2, 3, 4 | dih1cnv 38958 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (◡((DIsoH‘𝐾)‘𝑊)‘𝑉) = (1.‘𝐾)) |
12 | 11 | fveq2d 6691 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑉)) = ((oc‘𝐾)‘(1.‘𝐾))) |
13 | hlop 37032 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
14 | 13 | adantr 484 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
15 | eqid 2739 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
16 | 15, 10, 6 | opoc1 36872 | . . . . 5 ⊢ (𝐾 ∈ OP → ((oc‘𝐾)‘(1.‘𝐾)) = (0.‘𝐾)) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((oc‘𝐾)‘(1.‘𝐾)) = (0.‘𝐾)) |
18 | 12, 17 | eqtrd 2774 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑉)) = (0.‘𝐾)) |
19 | 18 | fveq2d 6691 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑉))) = (((DIsoH‘𝐾)‘𝑊)‘(0.‘𝐾))) |
20 | doch1.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
21 | 15, 1, 2, 3, 20 | dih0 38950 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoH‘𝐾)‘𝑊)‘(0.‘𝐾)) = { 0 }) |
22 | 9, 19, 21 | 3eqtrd 2778 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑉) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {csn 4526 ◡ccnv 5534 ran crn 5536 ‘cfv 6350 Basecbs 16599 occoc 16689 0gc0g 16829 0.cp0 17776 1.cp1 17777 OPcops 36842 HLchlt 37020 LHypclh 37654 DVecHcdvh 38748 DIsoHcdih 38898 ocHcoch 39017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-riotaBAD 36623 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-tpos 7934 df-undef 7981 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-n0 11990 df-z 12076 df-uz 12338 df-fz 12995 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-sca 16697 df-vsca 16698 df-0g 16831 df-proset 17667 df-poset 17685 df-plt 17697 df-lub 17713 df-glb 17714 df-join 17715 df-meet 17716 df-p0 17778 df-p1 17779 df-lat 17785 df-clat 17847 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-submnd 18086 df-grp 18235 df-minusg 18236 df-sbg 18237 df-subg 18407 df-cntz 18578 df-lsm 18892 df-cmn 19039 df-abl 19040 df-mgp 19372 df-ur 19384 df-ring 19431 df-oppr 19508 df-dvdsr 19526 df-unit 19527 df-invr 19557 df-dvr 19568 df-drng 19636 df-lmod 19768 df-lss 19836 df-lsp 19876 df-lvec 20007 df-oposet 36846 df-ol 36848 df-oml 36849 df-covers 36936 df-ats 36937 df-atl 36968 df-cvlat 36992 df-hlat 37021 df-llines 37168 df-lplanes 37169 df-lvols 37170 df-lines 37171 df-psubsp 37173 df-pmap 37174 df-padd 37466 df-lhyp 37658 df-laut 37659 df-ldil 37774 df-ltrn 37775 df-trl 37829 df-tendo 38425 df-edring 38427 df-disoa 38699 df-dvech 38749 df-dib 38809 df-dic 38843 df-dih 38899 df-doch 39018 |
This theorem is referenced by: dochoc0 39030 dochoc1 39031 dochn0nv 39045 djhexmid 39081 dochpolN 39160 lclkrs 39209 mapd0 39335 |
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