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Theorem ishlat1 37745
Description: The predicate "is a Hilbert lattice", which is: is orthomodular (𝐾 ∈ OML), complete (𝐾 ∈ CLat), atomic and satisfies the exchange (or covering) property (𝐾 ∈ CvLat), satisfies the superposition principle, and has a minimum height of 4 (witnessed here by 0, x, y, z, 1). (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
ishlat.b 𝐵 = (Base‘𝐾)
ishlat.l = (le‘𝐾)
ishlat.s < = (lt‘𝐾)
ishlat.j = (join‘𝐾)
ishlat.z 0 = (0.‘𝐾)
ishlat.u 1 = (1.‘𝐾)
ishlat.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
ishlat1 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧
Allowed substitution hints:   < (𝑥,𝑦,𝑧)   1 (𝑥,𝑦,𝑧)   (𝑥,𝑦,𝑧)   (𝑥,𝑦,𝑧)   0 (𝑥,𝑦,𝑧)

Proof of Theorem ishlat1
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6839 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2 ishlat.a . . . . . 6 𝐴 = (Atoms‘𝐾)
31, 2eqtr4di 2795 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4 fveq2 6839 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5 ishlat.l . . . . . . . . . . . 12 = (le‘𝐾)
64, 5eqtr4di 2795 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
76breqd 5114 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥(join‘𝑘)𝑦)))
8 fveq2 6839 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9 ishlat.j . . . . . . . . . . . . 13 = (join‘𝐾)
108, 9eqtr4di 2795 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
1110oveqd 7368 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑦) = (𝑥 𝑦))
1211breq2d 5115 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧 (𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥 𝑦)))
137, 12bitrd 278 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥 𝑦)))
14133anbi3d 1442 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦)) ↔ (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))))
153, 14rexeqbidv 3318 . . . . . . 7 (𝑘 = 𝐾 → (∃𝑧 ∈ (Atoms‘𝑘)(𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦)) ↔ ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))))
1615imbi2d 340 . . . . . 6 (𝑘 = 𝐾 → ((𝑥𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ↔ (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)))))
173, 16raleqbidv 3317 . . . . 5 (𝑘 = 𝐾 → (∀𝑦 ∈ (Atoms‘𝑘)(𝑥𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ↔ ∀𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)))))
183, 17raleqbidv 3317 . . . 4 (𝑘 = 𝐾 → (∀𝑥 ∈ (Atoms‘𝑘)∀𝑦 ∈ (Atoms‘𝑘)(𝑥𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)))))
19 fveq2 6839 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
20 ishlat.b . . . . . 6 𝐵 = (Base‘𝐾)
2119, 20eqtr4di 2795 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
22 fveq2 6839 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (lt‘𝑘) = (lt‘𝐾))
23 ishlat.s . . . . . . . . . . . 12 < = (lt‘𝐾)
2422, 23eqtr4di 2795 . . . . . . . . . . 11 (𝑘 = 𝐾 → (lt‘𝑘) = < )
2524breqd 5114 . . . . . . . . . 10 (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ (0.‘𝑘) < 𝑥))
26 fveq2 6839 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
27 ishlat.z . . . . . . . . . . . 12 0 = (0.‘𝐾)
2826, 27eqtr4di 2795 . . . . . . . . . . 11 (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
2928breq1d 5113 . . . . . . . . . 10 (𝑘 = 𝐾 → ((0.‘𝑘) < 𝑥0 < 𝑥))
3025, 29bitrd 278 . . . . . . . . 9 (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥0 < 𝑥))
3124breqd 5114 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥(lt‘𝑘)𝑦𝑥 < 𝑦))
3230, 31anbi12d 631 . . . . . . . 8 (𝑘 = 𝐾 → (((0.‘𝑘)(lt‘𝑘)𝑥𝑥(lt‘𝑘)𝑦) ↔ ( 0 < 𝑥𝑥 < 𝑦)))
3324breqd 5114 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑦(lt‘𝑘)𝑧𝑦 < 𝑧))
3424breqd 5114 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < (1.‘𝑘)))
35 fveq2 6839 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
36 ishlat.u . . . . . . . . . . . 12 1 = (1.‘𝐾)
3735, 36eqtr4di 2795 . . . . . . . . . . 11 (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
3837breq2d 5115 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧 < (1.‘𝑘) ↔ 𝑧 < 1 ))
3934, 38bitrd 278 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < 1 ))
4033, 39anbi12d 631 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑦(lt‘𝑘)𝑧𝑧(lt‘𝑘)(1.‘𝑘)) ↔ (𝑦 < 𝑧𝑧 < 1 )))
4132, 40anbi12d 631 . . . . . . 7 (𝑘 = 𝐾 → ((((0.‘𝑘)(lt‘𝑘)𝑥𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧𝑧(lt‘𝑘)(1.‘𝑘))) ↔ (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 ))))
4221, 41rexeqbidv 3318 . . . . . 6 (𝑘 = 𝐾 → (∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧𝑧(lt‘𝑘)(1.‘𝑘))) ↔ ∃𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 ))))
4321, 42rexeqbidv 3318 . . . . 5 (𝑘 = 𝐾 → (∃𝑦 ∈ (Base‘𝑘)∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧𝑧(lt‘𝑘)(1.‘𝑘))) ↔ ∃𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 ))))
4421, 43rexeqbidv 3318 . . . 4 (𝑘 = 𝐾 → (∃𝑥 ∈ (Base‘𝑘)∃𝑦 ∈ (Base‘𝑘)∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧𝑧(lt‘𝑘)(1.‘𝑘))) ↔ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 ))))
4518, 44anbi12d 631 . . 3 (𝑘 = 𝐾 → ((∀𝑥 ∈ (Atoms‘𝑘)∀𝑦 ∈ (Atoms‘𝑘)(𝑥𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ∧ ∃𝑥 ∈ (Base‘𝑘)∃𝑦 ∈ (Base‘𝑘)∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧𝑧(lt‘𝑘)(1.‘𝑘)))) ↔ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
46 df-hlat 37744 . . 3 HL = {𝑘 ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (∀𝑥 ∈ (Atoms‘𝑘)∀𝑦 ∈ (Atoms‘𝑘)(𝑥𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ∧ ∃𝑥 ∈ (Base‘𝑘)∃𝑦 ∈ (Base‘𝑘)∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧𝑧(lt‘𝑘)(1.‘𝑘))))}
4745, 46elrab2 3646 . 2 (𝐾 ∈ HL ↔ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
48 elin 3924 . . . . 5 (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
4948anbi1i 624 . . . 4 ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50 elin 3924 . . . 4 (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ↔ (𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat))
51 df-3an 1089 . . . 4 ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
5249, 50, 513bitr4ri 303 . . 3 ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ↔ 𝐾 ∈ ((OML ∩ CLat) ∩ CvLat))
5352anbi1i 624 . 2 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))) ↔ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
5447, 53bitr4i 277 1 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wral 3062  wrex 3071  cin 3907   class class class wbr 5103  cfv 6493  (class class class)co 7351  Basecbs 17042  lecple 17099  ltcplt 18156  joincjn 18159  0.cp0 18271  1.cp1 18272  CLatccla 18346  OMLcoml 37568  Atomscatm 37656  CvLatclc 37658  HLchlt 37743
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-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-ov 7354  df-hlat 37744
This theorem is referenced by:  ishlat2  37746  ishlat3N  37747  hlomcmcv  37749
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