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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochsnshp | Structured version Visualization version GIF version |
Description: The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.) |
Ref | Expression |
---|---|
dochsnshp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochsnshp.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochsnshp.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochsnshp.v | ⊢ 𝑉 = (Base‘𝑈) |
dochsnshp.z | ⊢ 0 = (0g‘𝑈) |
dochsnshp.y | ⊢ 𝑌 = (LSHyp‘𝑈) |
dochsnshp.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochsnshp.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
dochsnshp | ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochsnshp.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dochsnshp.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | dochsnshp.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | dochsnshp.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | eqid 2825 | . . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
6 | dochsnshp.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | dochsnshp.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
8 | 7 | eldifad 3810 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
9 | 8 | snssd 4558 | . . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
10 | 1, 2, 3, 4, 5, 6, 9 | dochocsp 37447 | . 2 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) = ( ⊥ ‘{𝑋})) |
11 | eqid 2825 | . . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
12 | dochsnshp.y | . . 3 ⊢ 𝑌 = (LSHyp‘𝑈) | |
13 | dochsnshp.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
14 | 1, 2, 6 | dvhlmod 37178 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
15 | 4, 5, 13, 11, 14, 7 | lsatlspsn 35061 | . . 3 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) ∈ (LSAtoms‘𝑈)) |
16 | 1, 2, 3, 11, 12, 6, 15 | dochsatshp 37519 | . 2 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) ∈ 𝑌) |
17 | 10, 16 | eqeltrrd 2907 | 1 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∖ cdif 3795 {csn 4397 ‘cfv 6123 Basecbs 16222 0gc0g 16453 LSpanclspn 19330 LSAtomsclsa 35042 LSHypclsh 35043 HLchlt 35418 LHypclh 36052 DVecHcdvh 37146 ocHcoch 37415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-riotaBAD 35021 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-undef 7664 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-0g 16455 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-p1 17393 df-lat 17399 df-clat 17461 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-dvr 19037 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lsatoms 35044 df-lshyp 35045 df-oposet 35244 df-ol 35246 df-oml 35247 df-covers 35334 df-ats 35335 df-atl 35366 df-cvlat 35390 df-hlat 35419 df-llines 35566 df-lplanes 35567 df-lvols 35568 df-lines 35569 df-psubsp 35571 df-pmap 35572 df-padd 35864 df-lhyp 36056 df-laut 36057 df-ldil 36172 df-ltrn 36173 df-trl 36227 df-tgrp 36811 df-tendo 36823 df-edring 36825 df-dveca 37071 df-disoa 37097 df-dvech 37147 df-dib 37207 df-dic 37241 df-dih 37297 df-doch 37416 df-djh 37463 |
This theorem is referenced by: dochexmidat 37527 dochsnkr2 37541 dochflcl 37543 dochfl1 37544 lcfl9a 37573 lclkrlem2a 37575 lcfrlem20 37630 lcfrlem25 37635 lcfrlem35 37645 hdmaplkr 37981 |
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