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Theorem ishlat1 39354
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 6905 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2 ishlat.a . . . . . 6 𝐴 = (Atoms‘𝐾)
31, 2eqtr4di 2794 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4 fveq2 6905 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5 ishlat.l . . . . . . . . . . . 12 = (le‘𝐾)
64, 5eqtr4di 2794 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
76breqd 5153 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥(join‘𝑘)𝑦)))
8 fveq2 6905 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9 ishlat.j . . . . . . . . . . . . 13 = (join‘𝐾)
108, 9eqtr4di 2794 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
1110oveqd 7449 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑦) = (𝑥 𝑦))
1211breq2d 5154 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧 (𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥 𝑦)))
137, 12bitrd 279 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥 𝑦)))
14133anbi3d 1443 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦)) ↔ (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))))
153, 14rexeqbidv 3346 . . . . . . 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 3345 . . . . 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 3345 . . . 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 6905 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
20 ishlat.b . . . . . 6 𝐵 = (Base‘𝐾)
2119, 20eqtr4di 2794 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
22 fveq2 6905 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (lt‘𝑘) = (lt‘𝐾))
23 ishlat.s . . . . . . . . . . . 12 < = (lt‘𝐾)
2422, 23eqtr4di 2794 . . . . . . . . . . 11 (𝑘 = 𝐾 → (lt‘𝑘) = < )
2524breqd 5153 . . . . . . . . . 10 (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ (0.‘𝑘) < 𝑥))
26 fveq2 6905 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
27 ishlat.z . . . . . . . . . . . 12 0 = (0.‘𝐾)
2826, 27eqtr4di 2794 . . . . . . . . . . 11 (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
2928breq1d 5152 . . . . . . . . . 10 (𝑘 = 𝐾 → ((0.‘𝑘) < 𝑥0 < 𝑥))
3025, 29bitrd 279 . . . . . . . . 9 (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥0 < 𝑥))
3124breqd 5153 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥(lt‘𝑘)𝑦𝑥 < 𝑦))
3230, 31anbi12d 632 . . . . . . . 8 (𝑘 = 𝐾 → (((0.‘𝑘)(lt‘𝑘)𝑥𝑥(lt‘𝑘)𝑦) ↔ ( 0 < 𝑥𝑥 < 𝑦)))
3324breqd 5153 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑦(lt‘𝑘)𝑧𝑦 < 𝑧))
3424breqd 5153 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < (1.‘𝑘)))
35 fveq2 6905 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
36 ishlat.u . . . . . . . . . . . 12 1 = (1.‘𝐾)
3735, 36eqtr4di 2794 . . . . . . . . . . 11 (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
3837breq2d 5154 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧 < (1.‘𝑘) ↔ 𝑧 < 1 ))
3934, 38bitrd 279 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < 1 ))
4033, 39anbi12d 632 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑦(lt‘𝑘)𝑧𝑧(lt‘𝑘)(1.‘𝑘)) ↔ (𝑦 < 𝑧𝑧 < 1 )))
4132, 40anbi12d 632 . . . . . . 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 3346 . . . . . 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 3346 . . . . 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 3346 . . . 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 632 . . 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 39353 . . 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 3694 . 2 (𝐾 ∈ HL ↔ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
48 elin 3966 . . . . 5 (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
4948anbi1i 624 . . . 4 ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50 elin 3966 . . . 4 (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ↔ (𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat))
51 df-3an 1088 . . . 4 ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
5249, 50, 513bitr4ri 304 . . 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 278 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 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  cin 3949   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  ltcplt 18355  joincjn 18358  0.cp0 18469  1.cp1 18470  CLatccla 18544  OMLcoml 39177  Atomscatm 39265  CvLatclc 39267  HLchlt 39352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-hlat 39353
This theorem is referenced by:  ishlat2  39355  ishlat3N  39356  hlomcmcv  39358
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