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

Theorem dochsatshp 39202
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 2737 . . . 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 38861 . . . 4 (𝜑𝑈 ∈ LMod)
7 dochsatshp.q . . . 4 (𝜑𝑄𝐴)
82, 3, 6, 7lsatssv 36749 . . 3 (𝜑𝑄 ⊆ (Base‘𝑈))
9 eqid 2737 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
10 dochsatshp.o . . . 4 = ((ocH‘𝐾)‘𝑊)
114, 5, 2, 9, 10dochlss 39105 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( 𝑄) ∈ (LSubSp‘𝑈))
121, 8, 11syl2anc 587 . 2 (𝜑 → ( 𝑄) ∈ (LSubSp‘𝑈))
13 eqid 2737 . . . 4 (0g𝑈) = (0g𝑈)
1413, 3, 6, 7lsatn0 36750 . . 3 (𝜑𝑄 ≠ {(0g𝑈)})
154, 5, 10, 2, 13doch0 39109 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ‘{(0g𝑈)}) = (Base‘𝑈))
161, 15syl 17 . . . . . 6 (𝜑 → ( ‘{(0g𝑈)}) = (Base‘𝑈))
1716eqeq2d 2748 . . . . 5 (𝜑 → (( 𝑄) = ( ‘{(0g𝑈)}) ↔ ( 𝑄) = (Base‘𝑈)))
18 eqid 2737 . . . . . 6 ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
194, 5, 18, 3dih1dimat 39081 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐴) → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
201, 7, 19syl2anc 587 . . . . . 6 (𝜑𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
214, 18, 5, 13dih0rn 39035 . . . . . . 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 39124 . . . . 5 (𝜑 → (( 𝑄) = ( ‘{(0g𝑈)}) ↔ 𝑄 = {(0g𝑈)}))
2417, 23bitr3d 284 . . . 4 (𝜑 → (( 𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g𝑈)}))
2524necon3bid 2985 . . 3 (𝜑 → (( 𝑄) ≠ (Base‘𝑈) ↔ 𝑄 ≠ {(0g𝑈)}))
2614, 25mpbird 260 . 2 (𝜑 → ( 𝑄) ≠ (Base‘𝑈))
27 eqid 2737 . . . . . 6 (LSpan‘𝑈) = (LSpan‘𝑈)
282, 27, 13, 3islsat 36742 . . . . 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 4041 . . . . . . 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 20019 . . . . . . . . . . . 12 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈)) → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
356, 12, 34syl2anc 587 . . . . . . . . . . 11 (𝜑 → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
3635uneq1d 4076 . . . . . . . . . 10 (𝜑 → (((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
3736fveq2d 6721 . . . . . . . . 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 19973 . . . . . . . . . . 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 4722 . . . . . . . . . . 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 20024 . . . . . . . . 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 1373 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))))
48 uneq2 4071 . . . . . . . . . . 11 (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( 𝑄) ∪ 𝑄) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
4948fveq2d 6721 . . . . . . . . . 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 2787 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
53 eqid 2737 . . . . . . . . . . 11 ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
54 eqid 2737 . . . . . . . . . . 11 (LSSum‘𝑈) = (LSSum‘𝑈)
554, 18, 5, 2, 10dochcl 39104 . . . . . . . . . . . 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 39182 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( 𝑄)(LSSum‘𝑈)𝑄))
584, 5, 2, 53, 1, 41, 8djhcom 39156 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)))
599, 3, 6, 7lsatlssel 36748 . . . . . . . . . . 11 (𝜑𝑄 ∈ (LSubSp‘𝑈))
609, 27, 54lsmsp 20123 . . . . . . . . . . 11 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈) ∧ 𝑄 ∈ (LSubSp‘𝑈)) → (( 𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
616, 12, 59, 60syl3anc 1373 . . . . . . . . . 10 (𝜑 → (( 𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
6257, 58, 613eqtr3rd 2786 . . . . . . . . 9 (𝜑 → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)))
634, 5, 2, 10, 53djhexmid 39162 . . . . . . . . . 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 2777 . . . . . . . 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 2777 . . . . . 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 3190 . . 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 38860 . . 3 (𝜑𝑈 ∈ LVec)
73 dochsatshp.y . . . 4 𝑌 = (LSHyp‘𝑈)
742, 27, 9, 73islshp 36730 . . 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 1344 1 (𝜑 → ( 𝑄) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wrex 3062  cdif 3863  cun 3864  wss 3866  {csn 4541  ran crn 5552  cfv 6380  (class class class)co 7213  Basecbs 16760  0gc0g 16944  LSSumclsm 19023  LModclmod 19899  LSubSpclss 19968  LSpanclspn 20008  LVecclvec 20139  LSAtomsclsa 36725  LSHypclsh 36726  HLchlt 37101  LHypclh 37735  DVecHcdvh 38829  DIsoHcdih 38979  ocHcoch 39098  joinHcdjh 39145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-riotaBAD 36704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-tpos 7968  df-undef 8015  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-sca 16818  df-vsca 16819  df-0g 16946  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-p1 17932  df-lat 17938  df-clat 18005  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-grp 18368  df-minusg 18369  df-sbg 18370  df-subg 18540  df-cntz 18711  df-lsm 19025  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-ring 19564  df-oppr 19641  df-dvdsr 19659  df-unit 19660  df-invr 19690  df-dvr 19701  df-drng 19769  df-lmod 19901  df-lss 19969  df-lsp 20009  df-lvec 20140  df-lsatoms 36727  df-lshyp 36728  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-llines 37249  df-lplanes 37250  df-lvols 37251  df-lines 37252  df-psubsp 37254  df-pmap 37255  df-padd 37547  df-lhyp 37739  df-laut 37740  df-ldil 37855  df-ltrn 37856  df-trl 37910  df-tgrp 38494  df-tendo 38506  df-edring 38508  df-dveca 38754  df-disoa 38780  df-dvech 38830  df-dib 38890  df-dic 38924  df-dih 38980  df-doch 39099  df-djh 39146
This theorem is referenced by:  dochsatshpb  39203  dochsnshp  39204  dochpolN  39241  lclkrlem2c  39260  lclkrlem2e  39262  mapdordlem2  39388
  Copyright terms: Public domain W3C validator