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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochspss | Structured version Visualization version GIF version |
Description: The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.) |
Ref | Expression |
---|---|
dochsp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochsp.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochsp.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochsp.v | ⊢ 𝑉 = (Base‘𝑈) |
dochsp.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dochsp.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochsp.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
Ref | Expression |
---|---|
dochspss | ⊢ (𝜑 → (𝑁‘𝑋) ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochsp.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dochsp.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | dochsp.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | eqid 2738 | . . . 4 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
5 | eqid 2738 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
6 | 2, 3, 4, 5 | dihsslss 38933 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran ((DIsoH‘𝐾)‘𝑊) ⊆ (LSubSp‘𝑈)) |
7 | rabss2 3967 | . . 3 ⊢ (ran ((DIsoH‘𝐾)‘𝑊) ⊆ (LSubSp‘𝑈) → {𝑧 ∈ ran ((DIsoH‘𝐾)‘𝑊) ∣ 𝑋 ⊆ 𝑧} ⊆ {𝑧 ∈ (LSubSp‘𝑈) ∣ 𝑋 ⊆ 𝑧}) | |
8 | intss 4857 | . . 3 ⊢ ({𝑧 ∈ ran ((DIsoH‘𝐾)‘𝑊) ∣ 𝑋 ⊆ 𝑧} ⊆ {𝑧 ∈ (LSubSp‘𝑈) ∣ 𝑋 ⊆ 𝑧} → ∩ {𝑧 ∈ (LSubSp‘𝑈) ∣ 𝑋 ⊆ 𝑧} ⊆ ∩ {𝑧 ∈ ran ((DIsoH‘𝐾)‘𝑊) ∣ 𝑋 ⊆ 𝑧}) | |
9 | 1, 6, 7, 8 | 4syl 19 | . 2 ⊢ (𝜑 → ∩ {𝑧 ∈ (LSubSp‘𝑈) ∣ 𝑋 ⊆ 𝑧} ⊆ ∩ {𝑧 ∈ ran ((DIsoH‘𝐾)‘𝑊) ∣ 𝑋 ⊆ 𝑧}) |
10 | 2, 3, 1 | dvhlmod 38767 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
11 | dochsp.x | . . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) | |
12 | dochsp.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
13 | dochsp.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
14 | 12, 5, 13 | lspval 19866 | . . 3 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = ∩ {𝑧 ∈ (LSubSp‘𝑈) ∣ 𝑋 ⊆ 𝑧}) |
15 | 10, 11, 14 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) = ∩ {𝑧 ∈ (LSubSp‘𝑈) ∣ 𝑋 ⊆ 𝑧}) |
16 | dochsp.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
17 | 2, 4, 3, 12, 16, 1, 11 | doch2val2 39021 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = ∩ {𝑧 ∈ ran ((DIsoH‘𝐾)‘𝑊) ∣ 𝑋 ⊆ 𝑧}) |
18 | 9, 15, 17 | 3sstr4d 3924 | 1 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {crab 3057 ⊆ wss 3843 ∩ cint 4836 ran crn 5526 ‘cfv 6339 Basecbs 16586 LModclmod 19753 LSubSpclss 19822 LSpanclspn 19862 HLchlt 37007 LHypclh 37641 DVecHcdvh 38735 DIsoHcdih 38885 ocHcoch 39004 |
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 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-riotaBAD 36610 |
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 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-undef 7968 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-0g 16818 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-p1 17766 df-lat 17772 df-clat 17834 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-cntz 18565 df-lsm 18879 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-drng 19623 df-lmod 19755 df-lss 19823 df-lsp 19863 df-lvec 19994 df-lsatoms 36633 df-oposet 36833 df-ol 36835 df-oml 36836 df-covers 36923 df-ats 36924 df-atl 36955 df-cvlat 36979 df-hlat 37008 df-llines 37155 df-lplanes 37156 df-lvols 37157 df-lines 37158 df-psubsp 37160 df-pmap 37161 df-padd 37453 df-lhyp 37645 df-laut 37646 df-ldil 37761 df-ltrn 37762 df-trl 37816 df-tendo 38412 df-edring 38414 df-disoa 38686 df-dvech 38736 df-dib 38796 df-dic 38830 df-dih 38886 df-doch 39005 |
This theorem is referenced by: dochocsp 39036 djhspss 39063 |
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