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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 2738 | . . . . . . 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 39364 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
9 | 1, 2, 8 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
10 | 3, 4, 7 | dochoc 39378 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
11 | 1, 9, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
12 | dochn0nv.z | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
13 | 3, 5, 7, 6, 12 | doch1 39370 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑉) = { 0 }) |
14 | 1, 13 | syl 17 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 }) |
15 | 11, 14 | eqeq12d 2754 | . . 3 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑉) ↔ ( ⊥ ‘𝑋) = { 0 })) |
16 | 3, 5, 6, 7 | dochssv 39366 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉) |
17 | 1, 2, 16 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉) |
18 | 3, 4, 5, 6, 7 | dochcl 39364 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
19 | 1, 17, 18 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
20 | 3, 4, 5, 6 | dih1rn 39298 | . . . . 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 39384 | . . 3 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑉) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑉)) |
23 | 15, 22 | bitr3d 280 | . 2 ⊢ (𝜑 → (( ⊥ ‘𝑋) = { 0 } ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑉)) |
24 | 23 | necon3bid 2988 | 1 ⊢ (𝜑 → (( ⊥ ‘𝑋) ≠ { 0 } ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ⊆ wss 3888 {csn 4563 ran crn 5592 ‘cfv 6435 Basecbs 16910 0gc0g 17148 HLchlt 37361 LHypclh 37995 DVecHcdvh 39089 DIsoHcdih 39239 ocHcoch 39358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-riotaBAD 36964 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8040 df-undef 8087 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-map 8615 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-n0 12232 df-z 12318 df-uz 12581 df-fz 13238 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-oposet 37187 df-ol 37189 df-oml 37190 df-covers 37277 df-ats 37278 df-atl 37309 df-cvlat 37333 df-hlat 37362 df-llines 37509 df-lplanes 37510 df-lvols 37511 df-lines 37512 df-psubsp 37514 df-pmap 37515 df-padd 37807 df-lhyp 37999 df-laut 38000 df-ldil 38115 df-ltrn 38116 df-trl 38170 df-tendo 38766 df-edring 38768 df-disoa 39040 df-dvech 39090 df-dib 39150 df-dic 39184 df-dih 39240 df-doch 39359 |
This theorem is referenced by: dochsnnz 39461 dochsatshpb 39463 dochkrsat 39466 dochkrsat2 39467 dochsnkrlem1 39480 |
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