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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochshpsat | Structured version Visualization version GIF version |
Description: A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.) |
Ref | Expression |
---|---|
dochshpsat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochshpsat.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochshpsat.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochshpsat.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
dochshpsat.y | ⊢ 𝑌 = (LSHyp‘𝑈) |
dochshpsat.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochshpsat.x | ⊢ (𝜑 → 𝑋 ∈ 𝑌) |
Ref | Expression |
---|---|
dochshpsat | ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) | |
2 | dochshpsat.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑌) | |
3 | 2 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ 𝑌) |
4 | 1, 3 | eqeltrd 2866 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌) |
5 | dochshpsat.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | dochshpsat.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
7 | dochshpsat.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | eqid 2778 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
9 | dochshpsat.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
10 | dochshpsat.y | . . . . 5 ⊢ 𝑌 = (LSHyp‘𝑈) | |
11 | dochshpsat.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | 5, 7, 11 | dvhlmod 37691 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
13 | 8, 10, 12, 2 | lshplss 35562 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (LSubSp‘𝑈)) |
14 | eqid 2778 | . . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
15 | 14, 8 | lssss 19433 | . . . . . . 7 ⊢ (𝑋 ∈ (LSubSp‘𝑈) → 𝑋 ⊆ (Base‘𝑈)) |
16 | 13, 15 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑈)) |
17 | 5, 7, 14, 8, 6 | dochlss 37935 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈)) |
18 | 11, 16, 17 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈)) |
19 | 5, 6, 7, 8, 9, 10, 11, 18 | dochsatshpb 38033 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌)) |
20 | 19 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌)) |
21 | 4, 20 | mpbird 249 | . 2 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ∈ 𝐴) |
22 | eqid 2778 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
23 | 12 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → 𝑈 ∈ LMod) |
24 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐴) | |
25 | 22, 9, 23, 24 | lsatn0 35580 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ≠ {(0g‘𝑈)}) |
26 | 25 | neneqd 2972 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘𝑋) = {(0g‘𝑈)}) |
27 | 11 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
28 | 5, 7, 6, 14, 22 | doch0 37939 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈)) |
29 | 27, 28 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈)) |
30 | 29 | eqeq2d 2788 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈))) |
31 | eqid 2778 | . . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
32 | 5, 31, 7, 14, 6 | dochcl 37934 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
33 | 11, 16, 32 | syl2anc 576 | . . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
34 | 5, 31, 7, 22 | dih0rn 37865 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {(0g‘𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
35 | 11, 34 | syl 17 | . . . . . . 7 ⊢ (𝜑 → {(0g‘𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
36 | 5, 31, 6, 11, 33, 35 | doch11 37954 | . . . . . 6 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0g‘𝑈)})) |
37 | 36 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0g‘𝑈)})) |
38 | 30, 37 | bitr3d 273 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘𝑋) = {(0g‘𝑈)})) |
39 | 26, 38 | mtbird 317 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈)) |
40 | 5, 6, 7, 14, 10, 11, 2 | dochshpncl 37965 | . . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑋 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈))) |
41 | 40 | necon1bbid 3006 | . . . 4 ⊢ (𝜑 → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
42 | 41 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
43 | 39, 42 | mpbid 224 | . 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
44 | 21, 43 | impbida 788 | 1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3831 {csn 4442 ran crn 5409 ‘cfv 6190 Basecbs 16342 0gc0g 16572 LModclmod 19359 LSubSpclss 19428 LSAtomsclsa 35555 LSHypclsh 35556 HLchlt 35931 LHypclh 36565 DVecHcdvh 37659 DIsoHcdih 37809 ocHcoch 37928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-riotaBAD 35534 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-tpos 7697 df-undef 7744 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-n0 11711 df-z 11797 df-uz 12062 df-fz 12712 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-sca 16440 df-vsca 16441 df-0g 16574 df-proset 17399 df-poset 17417 df-plt 17429 df-lub 17445 df-glb 17446 df-join 17447 df-meet 17448 df-p0 17510 df-p1 17511 df-lat 17517 df-clat 17579 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-grp 17897 df-minusg 17898 df-sbg 17899 df-subg 18063 df-cntz 18221 df-lsm 18525 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-oppr 19099 df-dvdsr 19117 df-unit 19118 df-invr 19148 df-dvr 19159 df-drng 19230 df-lmod 19361 df-lss 19429 df-lsp 19469 df-lvec 19600 df-lsatoms 35557 df-lshyp 35558 df-oposet 35757 df-ol 35759 df-oml 35760 df-covers 35847 df-ats 35848 df-atl 35879 df-cvlat 35903 df-hlat 35932 df-llines 36079 df-lplanes 36080 df-lvols 36081 df-lines 36082 df-psubsp 36084 df-pmap 36085 df-padd 36377 df-lhyp 36569 df-laut 36570 df-ldil 36685 df-ltrn 36686 df-trl 36740 df-tgrp 37324 df-tendo 37336 df-edring 37338 df-dveca 37584 df-disoa 37610 df-dvech 37660 df-dib 37720 df-dic 37754 df-dih 37810 df-doch 37929 df-djh 37976 |
This theorem is referenced by: mapdordlem2 38218 |
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