Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dochsatshp Structured version   Visualization version   GIF version

Theorem dochsatshp 38898
 Description: The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.)
Hypotheses
Ref Expression
dochsatshp.h 𝐻 = (LHyp‘𝐾)
dochsatshp.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochsatshp.o = ((ocH‘𝐾)‘𝑊)
dochsatshp.a 𝐴 = (LSAtoms‘𝑈)
dochsatshp.y 𝑌 = (LSHyp‘𝑈)
dochsatshp.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
dochsatshp.q (𝜑𝑄𝐴)
Assertion
Ref Expression
dochsatshp (𝜑 → ( 𝑄) ∈ 𝑌)

Proof of Theorem dochsatshp
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 dochsatshp.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 eqid 2798 . . . 4 (Base‘𝑈) = (Base‘𝑈)
3 dochsatshp.a . . . 4 𝐴 = (LSAtoms‘𝑈)
4 dochsatshp.h . . . . 5 𝐻 = (LHyp‘𝐾)
5 dochsatshp.u . . . . 5 𝑈 = ((DVecH‘𝐾)‘𝑊)
64, 5, 1dvhlmod 38557 . . . 4 (𝜑𝑈 ∈ LMod)
7 dochsatshp.q . . . 4 (𝜑𝑄𝐴)
82, 3, 6, 7lsatssv 36445 . . 3 (𝜑𝑄 ⊆ (Base‘𝑈))
9 eqid 2798 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
10 dochsatshp.o . . . 4 = ((ocH‘𝐾)‘𝑊)
114, 5, 2, 9, 10dochlss 38801 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( 𝑄) ∈ (LSubSp‘𝑈))
121, 8, 11syl2anc 587 . 2 (𝜑 → ( 𝑄) ∈ (LSubSp‘𝑈))
13 eqid 2798 . . . 4 (0g𝑈) = (0g𝑈)
1413, 3, 6, 7lsatn0 36446 . . 3 (𝜑𝑄 ≠ {(0g𝑈)})
154, 5, 10, 2, 13doch0 38805 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ‘{(0g𝑈)}) = (Base‘𝑈))
161, 15syl 17 . . . . . 6 (𝜑 → ( ‘{(0g𝑈)}) = (Base‘𝑈))
1716eqeq2d 2809 . . . . 5 (𝜑 → (( 𝑄) = ( ‘{(0g𝑈)}) ↔ ( 𝑄) = (Base‘𝑈)))
18 eqid 2798 . . . . . 6 ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
194, 5, 18, 3dih1dimat 38777 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐴) → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
201, 7, 19syl2anc 587 . . . . . 6 (𝜑𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
214, 18, 5, 13dih0rn 38731 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → {(0g𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊))
221, 21syl 17 . . . . . 6 (𝜑 → {(0g𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊))
234, 18, 10, 1, 20, 22doch11 38820 . . . . 5 (𝜑 → (( 𝑄) = ( ‘{(0g𝑈)}) ↔ 𝑄 = {(0g𝑈)}))
2417, 23bitr3d 284 . . . 4 (𝜑 → (( 𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g𝑈)}))
2524necon3bid 3031 . . 3 (𝜑 → (( 𝑄) ≠ (Base‘𝑈) ↔ 𝑄 ≠ {(0g𝑈)}))
2614, 25mpbird 260 . 2 (𝜑 → ( 𝑄) ≠ (Base‘𝑈))
27 eqid 2798 . . . . . 6 (LSpan‘𝑈) = (LSpan‘𝑈)
282, 27, 13, 3islsat 36438 . . . . 5 (𝑈 ∈ LMod → (𝑄𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣})))
296, 28syl 17 . . . 4 (𝜑 → (𝑄𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣})))
307, 29mpbid 235 . . 3 (𝜑 → ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}))
31 eldifi 4057 . . . . . . 7 (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) → 𝑣 ∈ (Base‘𝑈))
3231adantr 484 . . . . . 6 ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → 𝑣 ∈ (Base‘𝑈))
3332a1i 11 . . . . 5 (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → 𝑣 ∈ (Base‘𝑈)))
349, 27lspid 19768 . . . . . . . . . . . 12 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈)) → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
356, 12, 34syl2anc 587 . . . . . . . . . . 11 (𝜑 → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
3635uneq1d 4092 . . . . . . . . . 10 (𝜑 → (((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
3736fveq2d 6659 . . . . . . . . 9 (𝜑 → ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
3837adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
396adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → 𝑈 ∈ LMod)
402, 9lssss 19722 . . . . . . . . . . 11 (( 𝑄) ∈ (LSubSp‘𝑈) → ( 𝑄) ⊆ (Base‘𝑈))
4112, 40syl 17 . . . . . . . . . 10 (𝜑 → ( 𝑄) ⊆ (Base‘𝑈))
4241adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ( 𝑄) ⊆ (Base‘𝑈))
4331snssd 4705 . . . . . . . . . . 11 (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) → {𝑣} ⊆ (Base‘𝑈))
4443adantr 484 . . . . . . . . . 10 ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → {𝑣} ⊆ (Base‘𝑈))
4544adantl 485 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → {𝑣} ⊆ (Base‘𝑈))
462, 27lspun 19773 . . . . . . . . 9 ((𝑈 ∈ LMod ∧ ( 𝑄) ⊆ (Base‘𝑈) ∧ {𝑣} ⊆ (Base‘𝑈)) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))))
4739, 42, 45, 46syl3anc 1368 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))))
48 uneq2 4087 . . . . . . . . . . 11 (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( 𝑄) ∪ 𝑄) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
4948fveq2d 6659 . . . . . . . . . 10 (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
5049adantl 485 . . . . . . . . 9 ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
5150adantl 485 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
5238, 47, 513eqtr4d 2843 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
53 eqid 2798 . . . . . . . . . . 11 ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
54 eqid 2798 . . . . . . . . . . 11 (LSSum‘𝑈) = (LSSum‘𝑈)
554, 18, 5, 2, 10dochcl 38800 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
561, 8, 55syl2anc 587 . . . . . . . . . . 11 (𝜑 → ( 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
574, 18, 53, 5, 54, 3, 1, 56, 7dihjat2 38878 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( 𝑄)(LSSum‘𝑈)𝑄))
584, 5, 2, 53, 1, 41, 8djhcom 38852 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)))
599, 3, 6, 7lsatlssel 36444 . . . . . . . . . . 11 (𝜑𝑄 ∈ (LSubSp‘𝑈))
609, 27, 54lsmsp 19872 . . . . . . . . . . 11 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈) ∧ 𝑄 ∈ (LSubSp‘𝑈)) → (( 𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
616, 12, 59, 60syl3anc 1368 . . . . . . . . . 10 (𝜑 → (( 𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
6257, 58, 613eqtr3rd 2842 . . . . . . . . 9 (𝜑 → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)))
634, 5, 2, 10, 53djhexmid 38858 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)) = (Base‘𝑈))
641, 8, 63syl2anc 587 . . . . . . . . 9 (𝜑 → (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)) = (Base‘𝑈))
6562, 64eqtrd 2833 . . . . . . . 8 (𝜑 → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (Base‘𝑈))
6665adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (Base‘𝑈))
6752, 66eqtrd 2833 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈))
6867ex 416 . . . . 5 (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈)))
6933, 68jcad 516 . . . 4 (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → (𝑣 ∈ (Base‘𝑈) ∧ ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈))))
7069reximdv2 3231 . . 3 (𝜑 → (∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}) → ∃𝑣 ∈ (Base‘𝑈)((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈)))
7130, 70mpd 15 . 2 (𝜑 → ∃𝑣 ∈ (Base‘𝑈)((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈))
724, 5, 1dvhlvec 38556 . . 3 (𝜑𝑈 ∈ LVec)
73 dochsatshp.y . . . 4 𝑌 = (LSHyp‘𝑈)
742, 27, 9, 73islshp 36426 . . 3 (𝑈 ∈ LVec → (( 𝑄) ∈ 𝑌 ↔ (( 𝑄) ∈ (LSubSp‘𝑈) ∧ ( 𝑄) ≠ (Base‘𝑈) ∧ ∃𝑣 ∈ (Base‘𝑈)((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈))))
7572, 74syl 17 . 2 (𝜑 → (( 𝑄) ∈ 𝑌 ↔ (( 𝑄) ∈ (LSubSp‘𝑈) ∧ ( 𝑄) ≠ (Base‘𝑈) ∧ ∃𝑣 ∈ (Base‘𝑈)((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈))))
7612, 26, 71, 75mpbir3and 1339 1 (𝜑 → ( 𝑄) ∈ 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  {csn 4528  ran crn 5524  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  0gc0g 16725  LSSumclsm 18772  LModclmod 19648  LSubSpclss 19717  LSpanclspn 19757  LVecclvec 19888  LSAtomsclsa 36421  LSHypclsh 36422  HLchlt 36797  LHypclh 37431  DVecHcdvh 38525  DIsoHcdih 38675  ocHcoch 38794  joinHcdjh 38841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-riotaBAD 36400 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-tpos 7893  df-undef 7940  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-map 8409  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-3 11707  df-4 11708  df-5 11709  df-6 11710  df-n0 11904  df-z 11990  df-uz 12252  df-fz 12906  df-struct 16497  df-ndx 16498  df-slot 16499  df-base 16501  df-sets 16502  df-ress 16503  df-plusg 16590  df-mulr 16591  df-sca 16593  df-vsca 16594  df-0g 16727  df-proset 17550  df-poset 17568  df-plt 17580  df-lub 17596  df-glb 17597  df-join 17598  df-meet 17599  df-p0 17661  df-p1 17662  df-lat 17668  df-clat 17730  df-mgm 17864  df-sgrp 17913  df-mnd 17924  df-submnd 17969  df-grp 18118  df-minusg 18119  df-sbg 18120  df-subg 18289  df-cntz 18460  df-lsm 18774  df-cmn 18921  df-abl 18922  df-mgp 19254  df-ur 19266  df-ring 19313  df-oppr 19390  df-dvdsr 19408  df-unit 19409  df-invr 19439  df-dvr 19450  df-drng 19518  df-lmod 19650  df-lss 19718  df-lsp 19758  df-lvec 19889  df-lsatoms 36423  df-lshyp 36424  df-oposet 36623  df-ol 36625  df-oml 36626  df-covers 36713  df-ats 36714  df-atl 36745  df-cvlat 36769  df-hlat 36798  df-llines 36945  df-lplanes 36946  df-lvols 36947  df-lines 36948  df-psubsp 36950  df-pmap 36951  df-padd 37243  df-lhyp 37435  df-laut 37436  df-ldil 37551  df-ltrn 37552  df-trl 37606  df-tgrp 38190  df-tendo 38202  df-edring 38204  df-dveca 38450  df-disoa 38476  df-dvech 38526  df-dib 38586  df-dic 38620  df-dih 38676  df-doch 38795  df-djh 38842 This theorem is referenced by:  dochsatshpb  38899  dochsnshp  38900  dochpolN  38937  lclkrlem2c  38956  lclkrlem2e  38958  mapdordlem2  39084
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