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 39914
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 39573 . . . 4 (𝜑𝑈 ∈ LMod)
7 dochsatshp.q . . . 4 (𝜑𝑄𝐴)
82, 3, 6, 7lsatssv 37460 . . 3 (𝜑𝑄 ⊆ (Base‘𝑈))
9 eqid 2736 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
10 dochsatshp.o . . . 4 = ((ocH‘𝐾)‘𝑊)
114, 5, 2, 9, 10dochlss 39817 . . 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 37461 . . 3 (𝜑𝑄 ≠ {(0g𝑈)})
154, 5, 10, 2, 13doch0 39821 . . . . . . 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 39793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐴) → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
201, 7, 19syl2anc 584 . . . . . 6 (𝜑𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
214, 18, 5, 13dih0rn 39747 . . . . . . 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 39836 . . . . 5 (𝜑 → (( 𝑄) = ( ‘{(0g𝑈)}) ↔ 𝑄 = {(0g𝑈)}))
2417, 23bitr3d 280 . . . 4 (𝜑 → (( 𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g𝑈)}))
2524necon3bid 2988 . . 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 37453 . . . . 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 4086 . . . . . . 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 20443 . . . . . . . . . . . 12 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈)) → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
356, 12, 34syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
3635uneq1d 4122 . . . . . . . . . 10 (𝜑 → (((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
3736fveq2d 6846 . . . . . . . . 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 20397 . . . . . . . . . . 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 4769 . . . . . . . . . . 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 20448 . . . . . . . . 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 1371 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))))
48 uneq2 4117 . . . . . . . . . . 11 (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( 𝑄) ∪ 𝑄) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
4948fveq2d 6846 . . . . . . . . . 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 39816 . . . . . . . . . . . 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 39894 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( 𝑄)(LSSum‘𝑈)𝑄))
584, 5, 2, 53, 1, 41, 8djhcom 39868 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)))
599, 3, 6, 7lsatlssel 37459 . . . . . . . . . . 11 (𝜑𝑄 ∈ (LSubSp‘𝑈))
609, 27, 54lsmsp 20547 . . . . . . . . . . 11 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈) ∧ 𝑄 ∈ (LSubSp‘𝑈)) → (( 𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
616, 12, 59, 60syl3anc 1371 . . . . . . . . . 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 39874 . . . . . . . . . 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 3161 . . 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 39572 . . 3 (𝜑𝑈 ∈ LVec)
73 dochsatshp.y . . . 4 𝑌 = (LSHyp‘𝑈)
742, 27, 9, 73islshp 37441 . . 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 1342 1 (𝜑 → ( 𝑄) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073  cdif 3907  cun 3908  wss 3910  {csn 4586  ran crn 5634  cfv 6496  (class class class)co 7357  Basecbs 17083  0gc0g 17321  LSSumclsm 19416  LModclmod 20322  LSubSpclss 20392  LSpanclspn 20432  LVecclvec 20563  LSAtomsclsa 37436  LSHypclsh 37437  HLchlt 37812  LHypclh 38447  DVecHcdvh 39541  DIsoHcdih 39691  ocHcoch 39810  joinHcdjh 39857
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-riotaBAD 37415
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-undef 8204  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-0g 17323  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-cntz 19097  df-lsm 19418  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-lmod 20324  df-lss 20393  df-lsp 20433  df-lvec 20564  df-lsatoms 37438  df-lshyp 37439  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622  df-tgrp 39206  df-tendo 39218  df-edring 39220  df-dveca 39466  df-disoa 39492  df-dvech 39542  df-dib 39602  df-dic 39636  df-dih 39692  df-doch 39811  df-djh 39858
This theorem is referenced by:  dochsatshpb  39915  dochsnshp  39916  dochpolN  39953  lclkrlem2c  39972  lclkrlem2e  39974  mapdordlem2  40100
  Copyright terms: Public domain W3C validator