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Theorem dochsatshp 39719
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 2736 . . . 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 39378 . . . 4 (𝜑𝑈 ∈ LMod)
7 dochsatshp.q . . . 4 (𝜑𝑄𝐴)
82, 3, 6, 7lsatssv 37265 . . 3 (𝜑𝑄 ⊆ (Base‘𝑈))
9 eqid 2736 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
10 dochsatshp.o . . . 4 = ((ocH‘𝐾)‘𝑊)
114, 5, 2, 9, 10dochlss 39622 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( 𝑄) ∈ (LSubSp‘𝑈))
121, 8, 11syl2anc 584 . 2 (𝜑 → ( 𝑄) ∈ (LSubSp‘𝑈))
13 eqid 2736 . . . 4 (0g𝑈) = (0g𝑈)
1413, 3, 6, 7lsatn0 37266 . . 3 (𝜑𝑄 ≠ {(0g𝑈)})
154, 5, 10, 2, 13doch0 39626 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ‘{(0g𝑈)}) = (Base‘𝑈))
161, 15syl 17 . . . . . 6 (𝜑 → ( ‘{(0g𝑈)}) = (Base‘𝑈))
1716eqeq2d 2747 . . . . 5 (𝜑 → (( 𝑄) = ( ‘{(0g𝑈)}) ↔ ( 𝑄) = (Base‘𝑈)))
18 eqid 2736 . . . . . 6 ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
194, 5, 18, 3dih1dimat 39598 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐴) → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
201, 7, 19syl2anc 584 . . . . . 6 (𝜑𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
214, 18, 5, 13dih0rn 39552 . . . . . . 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 39641 . . . . 5 (𝜑 → (( 𝑄) = ( ‘{(0g𝑈)}) ↔ 𝑄 = {(0g𝑈)}))
2417, 23bitr3d 280 . . . 4 (𝜑 → (( 𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g𝑈)}))
2524necon3bid 2985 . . 3 (𝜑 → (( 𝑄) ≠ (Base‘𝑈) ↔ 𝑄 ≠ {(0g𝑈)}))
2614, 25mpbird 256 . 2 (𝜑 → ( 𝑄) ≠ (Base‘𝑈))
27 eqid 2736 . . . . . 6 (LSpan‘𝑈) = (LSpan‘𝑈)
282, 27, 13, 3islsat 37258 . . . . 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 231 . . 3 (𝜑 → ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}))
31 eldifi 4073 . . . . . . 7 (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) → 𝑣 ∈ (Base‘𝑈))
3231adantr 481 . . . . . 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 20350 . . . . . . . . . . . 12 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈)) → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
356, 12, 34syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
3635uneq1d 4109 . . . . . . . . . 10 (𝜑 → (((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
3736fveq2d 6829 . . . . . . . . 9 (𝜑 → ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
3837adantr 481 . . . . . . . 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 481 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → 𝑈 ∈ LMod)
402, 9lssss 20304 . . . . . . . . . . 11 (( 𝑄) ∈ (LSubSp‘𝑈) → ( 𝑄) ⊆ (Base‘𝑈))
4112, 40syl 17 . . . . . . . . . 10 (𝜑 → ( 𝑄) ⊆ (Base‘𝑈))
4241adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ( 𝑄) ⊆ (Base‘𝑈))
4331snssd 4756 . . . . . . . . . . 11 (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) → {𝑣} ⊆ (Base‘𝑈))
4443adantr 481 . . . . . . . . . 10 ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → {𝑣} ⊆ (Base‘𝑈))
4544adantl 482 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → {𝑣} ⊆ (Base‘𝑈))
462, 27lspun 20355 . . . . . . . . 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 1370 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))))
48 uneq2 4104 . . . . . . . . . . 11 (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( 𝑄) ∪ 𝑄) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
4948fveq2d 6829 . . . . . . . . . 10 (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
5049adantl 482 . . . . . . . . 9 ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
5150adantl 482 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
5238, 47, 513eqtr4d 2786 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
53 eqid 2736 . . . . . . . . . . 11 ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
54 eqid 2736 . . . . . . . . . . 11 (LSSum‘𝑈) = (LSSum‘𝑈)
554, 18, 5, 2, 10dochcl 39621 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
561, 8, 55syl2anc 584 . . . . . . . . . . 11 (𝜑 → ( 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
574, 18, 53, 5, 54, 3, 1, 56, 7dihjat2 39699 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( 𝑄)(LSSum‘𝑈)𝑄))
584, 5, 2, 53, 1, 41, 8djhcom 39673 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)))
599, 3, 6, 7lsatlssel 37264 . . . . . . . . . . 11 (𝜑𝑄 ∈ (LSubSp‘𝑈))
609, 27, 54lsmsp 20454 . . . . . . . . . . 11 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈) ∧ 𝑄 ∈ (LSubSp‘𝑈)) → (( 𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
616, 12, 59, 60syl3anc 1370 . . . . . . . . . 10 (𝜑 → (( 𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
6257, 58, 613eqtr3rd 2785 . . . . . . . . 9 (𝜑 → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)))
634, 5, 2, 10, 53djhexmid 39679 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)) = (Base‘𝑈))
641, 8, 63syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)) = (Base‘𝑈))
6562, 64eqtrd 2776 . . . . . . . 8 (𝜑 → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (Base‘𝑈))
6665adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (Base‘𝑈))
6752, 66eqtrd 2776 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈))
6867ex 413 . . . . 5 (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈)))
6933, 68jcad 513 . . . 4 (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → (𝑣 ∈ (Base‘𝑈) ∧ ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈))))
7069reximdv2 3157 . . 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 39377 . . 3 (𝜑𝑈 ∈ LVec)
73 dochsatshp.y . . . 4 𝑌 = (LSHyp‘𝑈)
742, 27, 9, 73islshp 37246 . . 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 1341 1 (𝜑 → ( 𝑄) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wne 2940  wrex 3070  cdif 3895  cun 3896  wss 3898  {csn 4573  ran crn 5621  cfv 6479  (class class class)co 7337  Basecbs 17009  0gc0g 17247  LSSumclsm 19335  LModclmod 20229  LSubSpclss 20299  LSpanclspn 20339  LVecclvec 20470  LSAtomsclsa 37241  LSHypclsh 37242  HLchlt 37617  LHypclh 38252  DVecHcdvh 39346  DIsoHcdih 39496  ocHcoch 39615  joinHcdjh 39662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-riotaBAD 37220
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-tpos 8112  df-undef 8159  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-map 8688  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-n0 12335  df-z 12421  df-uz 12684  df-fz 13341  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-mulr 17073  df-sca 17075  df-vsca 17076  df-0g 17249  df-proset 18110  df-poset 18128  df-plt 18145  df-lub 18161  df-glb 18162  df-join 18163  df-meet 18164  df-p0 18240  df-p1 18241  df-lat 18247  df-clat 18314  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-submnd 18528  df-grp 18676  df-minusg 18677  df-sbg 18678  df-subg 18848  df-cntz 19019  df-lsm 19337  df-cmn 19483  df-abl 19484  df-mgp 19816  df-ur 19833  df-ring 19880  df-oppr 19957  df-dvdsr 19978  df-unit 19979  df-invr 20009  df-dvr 20020  df-drng 20095  df-lmod 20231  df-lss 20300  df-lsp 20340  df-lvec 20471  df-lsatoms 37243  df-lshyp 37244  df-oposet 37443  df-ol 37445  df-oml 37446  df-covers 37533  df-ats 37534  df-atl 37565  df-cvlat 37589  df-hlat 37618  df-llines 37766  df-lplanes 37767  df-lvols 37768  df-lines 37769  df-psubsp 37771  df-pmap 37772  df-padd 38064  df-lhyp 38256  df-laut 38257  df-ldil 38372  df-ltrn 38373  df-trl 38427  df-tgrp 39011  df-tendo 39023  df-edring 39025  df-dveca 39271  df-disoa 39297  df-dvech 39347  df-dib 39407  df-dic 39441  df-dih 39497  df-doch 39616  df-djh 39663
This theorem is referenced by:  dochsatshpb  39720  dochsnshp  39721  dochpolN  39758  lclkrlem2c  39777  lclkrlem2e  39779  mapdordlem2  39905
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