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Theorem dochsatshp 37254
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 2771 . . . 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 36913 . . . 4 (𝜑𝑈 ∈ LMod)
7 dochsatshp.q . . . 4 (𝜑𝑄𝐴)
82, 3, 6, 7lsatssv 34800 . . 3 (𝜑𝑄 ⊆ (Base‘𝑈))
9 eqid 2771 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
10 dochsatshp.o . . . 4 = ((ocH‘𝐾)‘𝑊)
114, 5, 2, 9, 10dochlss 37157 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( 𝑄) ∈ (LSubSp‘𝑈))
121, 8, 11syl2anc 573 . 2 (𝜑 → ( 𝑄) ∈ (LSubSp‘𝑈))
13 eqid 2771 . . . 4 (0g𝑈) = (0g𝑈)
1413, 3, 6, 7lsatn0 34801 . . 3 (𝜑𝑄 ≠ {(0g𝑈)})
154, 5, 10, 2, 13doch0 37161 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ‘{(0g𝑈)}) = (Base‘𝑈))
161, 15syl 17 . . . . . 6 (𝜑 → ( ‘{(0g𝑈)}) = (Base‘𝑈))
1716eqeq2d 2781 . . . . 5 (𝜑 → (( 𝑄) = ( ‘{(0g𝑈)}) ↔ ( 𝑄) = (Base‘𝑈)))
18 eqid 2771 . . . . . 6 ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
194, 5, 18, 3dih1dimat 37133 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐴) → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
201, 7, 19syl2anc 573 . . . . . 6 (𝜑𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
214, 18, 5, 13dih0rn 37087 . . . . . . 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 37176 . . . . 5 (𝜑 → (( 𝑄) = ( ‘{(0g𝑈)}) ↔ 𝑄 = {(0g𝑈)}))
2417, 23bitr3d 270 . . . 4 (𝜑 → (( 𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g𝑈)}))
2524necon3bid 2987 . . 3 (𝜑 → (( 𝑄) ≠ (Base‘𝑈) ↔ 𝑄 ≠ {(0g𝑈)}))
2614, 25mpbird 247 . 2 (𝜑 → ( 𝑄) ≠ (Base‘𝑈))
27 eqid 2771 . . . . . 6 (LSpan‘𝑈) = (LSpan‘𝑈)
282, 27, 13, 3islsat 34793 . . . . 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 222 . . 3 (𝜑 → ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}))
31 eldifi 3883 . . . . . . 7 (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) → 𝑣 ∈ (Base‘𝑈))
3231adantr 466 . . . . . 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 19188 . . . . . . . . . . . 12 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈)) → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
356, 12, 34syl2anc 573 . . . . . . . . . . 11 (𝜑 → ((LSpan‘𝑈)‘( 𝑄)) = ( 𝑄))
3635uneq1d 3917 . . . . . . . . . 10 (𝜑 → (((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
3736fveq2d 6334 . . . . . . . . 9 (𝜑 → ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
3837adantr 466 . . . . . . . 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 466 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → 𝑈 ∈ LMod)
402, 9lssss 19140 . . . . . . . . . . 11 (( 𝑄) ∈ (LSubSp‘𝑈) → ( 𝑄) ⊆ (Base‘𝑈))
4112, 40syl 17 . . . . . . . . . 10 (𝜑 → ( 𝑄) ⊆ (Base‘𝑈))
4241adantr 466 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ( 𝑄) ⊆ (Base‘𝑈))
4331snssd 4475 . . . . . . . . . . 11 (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) → {𝑣} ⊆ (Base‘𝑈))
4443adantr 466 . . . . . . . . . 10 ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → {𝑣} ⊆ (Base‘𝑈))
4544adantl 467 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → {𝑣} ⊆ (Base‘𝑈))
462, 27lspun 19193 . . . . . . . . 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 1476 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( 𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))))
48 uneq2 3912 . . . . . . . . . . 11 (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( 𝑄) ∪ 𝑄) = (( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
4948fveq2d 6334 . . . . . . . . . 10 (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
5049adantl 467 . . . . . . . . 9 ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
5150adantl 467 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( 𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
5238, 47, 513eqtr4d 2815 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
53 eqid 2771 . . . . . . . . . . 11 ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
54 eqid 2771 . . . . . . . . . . 11 (LSSum‘𝑈) = (LSSum‘𝑈)
554, 18, 5, 2, 10dochcl 37156 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
561, 8, 55syl2anc 573 . . . . . . . . . . 11 (𝜑 → ( 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
574, 18, 53, 5, 54, 3, 1, 56, 7dihjat2 37234 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( 𝑄)(LSSum‘𝑈)𝑄))
584, 5, 2, 53, 1, 41, 8djhcom 37208 . . . . . . . . . 10 (𝜑 → (( 𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)))
599, 3, 6, 7lsatlssel 34799 . . . . . . . . . . 11 (𝜑𝑄 ∈ (LSubSp‘𝑈))
609, 27, 54lsmsp 19292 . . . . . . . . . . 11 ((𝑈 ∈ LMod ∧ ( 𝑄) ∈ (LSubSp‘𝑈) ∧ 𝑄 ∈ (LSubSp‘𝑈)) → (( 𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
616, 12, 59, 60syl3anc 1476 . . . . . . . . . 10 (𝜑 → (( 𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)))
6257, 58, 613eqtr3rd 2814 . . . . . . . . 9 (𝜑 → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)))
634, 5, 2, 10, 53djhexmid 37214 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)) = (Base‘𝑈))
641, 8, 63syl2anc 573 . . . . . . . . 9 (𝜑 → (𝑄((joinH‘𝐾)‘𝑊)( 𝑄)) = (Base‘𝑈))
6562, 64eqtrd 2805 . . . . . . . 8 (𝜑 → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (Base‘𝑈))
6665adantr 466 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ 𝑄)) = (Base‘𝑈))
6752, 66eqtrd 2805 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈))
6867ex 397 . . . . 5 (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈)))
6933, 68jcad 502 . . . 4 (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → (𝑣 ∈ (Base‘𝑈) ∧ ((LSpan‘𝑈)‘(( 𝑄) ∪ {𝑣})) = (Base‘𝑈))))
7069reximdv2 3162 . . 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 36912 . . 3 (𝜑𝑈 ∈ LVec)
73 dochsatshp.y . . . 4 𝑌 = (LSHyp‘𝑈)
742, 27, 9, 73islshp 34781 . . 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 1427 1 (𝜑 → ( 𝑄) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062  cdif 3720  cun 3721  wss 3723  {csn 4316  ran crn 5250  cfv 6029  (class class class)co 6791  Basecbs 16057  0gc0g 16301  LSSumclsm 18249  LModclmod 19066  LSubSpclss 19135  LSpanclspn 19177  LVecclvec 19308  LSAtomsclsa 34776  LSHypclsh 34777  HLchlt 35152  LHypclh 35785  DVecHcdvh 36881  DIsoHcdih 37031  ocHcoch 37150  joinHcdjh 37197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213  ax-riotaBAD 34754
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-tpos 7502  df-undef 7549  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-2 11279  df-3 11280  df-4 11281  df-5 11282  df-6 11283  df-n0 11493  df-z 11578  df-uz 11887  df-fz 12527  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16155  df-mulr 16156  df-sca 16158  df-vsca 16159  df-0g 16303  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-p1 17241  df-lat 17247  df-clat 17309  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-subg 17792  df-cntz 17950  df-lsm 18251  df-cmn 18395  df-abl 18396  df-mgp 18691  df-ur 18703  df-ring 18750  df-oppr 18824  df-dvdsr 18842  df-unit 18843  df-invr 18873  df-dvr 18884  df-drng 18952  df-lmod 19068  df-lss 19136  df-lsp 19178  df-lvec 19309  df-lsatoms 34778  df-lshyp 34779  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300  df-lvols 35301  df-lines 35302  df-psubsp 35304  df-pmap 35305  df-padd 35597  df-lhyp 35789  df-laut 35790  df-ldil 35905  df-ltrn 35906  df-trl 35961  df-tgrp 36545  df-tendo 36557  df-edring 36559  df-dveca 36805  df-disoa 36832  df-dvech 36882  df-dib 36942  df-dic 36976  df-dih 37032  df-doch 37151  df-djh 37198
This theorem is referenced by:  dochsatshpb  37255  dochsnshp  37256  dochpolN  37293  lclkrlem2c  37312  lclkrlem2e  37314  mapdordlem2  37440
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