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Theorem ishlat1 37814
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 6842 . . . . . 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 6842 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5 ishlat.l . . . . . . . . . . . 12 = (le‘𝐾)
64, 5eqtr4di 2794 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
76breqd 5116 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥(join‘𝑘)𝑦)))
8 fveq2 6842 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9 ishlat.j . . . . . . . . . . . . 13 = (join‘𝐾)
108, 9eqtr4di 2794 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
1110oveqd 7374 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑦) = (𝑥 𝑦))
1211breq2d 5117 . . . . . . . . . 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 3320 . . . . . . 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 3319 . . . . 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 3319 . . . 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 6842 . . . . . 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 6842 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (lt‘𝑘) = (lt‘𝐾))
23 ishlat.s . . . . . . . . . . . 12 < = (lt‘𝐾)
2422, 23eqtr4di 2794 . . . . . . . . . . 11 (𝑘 = 𝐾 → (lt‘𝑘) = < )
2524breqd 5116 . . . . . . . . . 10 (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ (0.‘𝑘) < 𝑥))
26 fveq2 6842 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
27 ishlat.z . . . . . . . . . . . 12 0 = (0.‘𝐾)
2826, 27eqtr4di 2794 . . . . . . . . . . 11 (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
2928breq1d 5115 . . . . . . . . . 10 (𝑘 = 𝐾 → ((0.‘𝑘) < 𝑥0 < 𝑥))
3025, 29bitrd 278 . . . . . . . . 9 (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥0 < 𝑥))
3124breqd 5116 . . . . . . . . 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 5116 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑦(lt‘𝑘)𝑧𝑦 < 𝑧))
3424breqd 5116 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < (1.‘𝑘)))
35 fveq2 6842 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
36 ishlat.u . . . . . . . . . . . 12 1 = (1.‘𝐾)
3735, 36eqtr4di 2794 . . . . . . . . . . 11 (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
3837breq2d 5117 . . . . . . . . . 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 3320 . . . . . 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 3320 . . . . 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 3320 . . . 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 37813 . . 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 3648 . 2 (𝐾 ∈ HL ↔ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
48 elin 3926 . . . . 5 (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
4948anbi1i 624 . . . 4 ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50 elin 3926 . . . 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 2943  wral 3064  wrex 3073  cin 3909   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  ltcplt 18197  joincjn 18200  0.cp0 18312  1.cp1 18313  CLatccla 18387  OMLcoml 37637  Atomscatm 37725  CvLatclc 37727  HLchlt 37812
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 2707
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 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360  df-hlat 37813
This theorem is referenced by:  ishlat2  37815  ishlat3N  37816  hlomcmcv  37818
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