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Theorem ishlat1 35160
Description: The predicate "is a Hilbert lattice," which is orthomodular (𝐾 ∈ OML), complete (𝐾 ∈ CLat), atomic and satisfying the exchange (or covering) property (𝐾 ∈ CvLat), satisfies the superposition principle, and has a minimum height of 4. (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 6353 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2 ishlat.a . . . . . 6 𝐴 = (Atoms‘𝐾)
31, 2syl6eqr 2812 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4 fveq2 6353 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5 ishlat.l . . . . . . . . . . . 12 = (le‘𝐾)
64, 5syl6eqr 2812 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = )
76breqd 4815 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥(join‘𝑘)𝑦)))
8 fveq2 6353 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9 ishlat.j . . . . . . . . . . . . 13 = (join‘𝐾)
108, 9syl6eqr 2812 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
1110oveqd 6831 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑦) = (𝑥 𝑦))
1211breq2d 4816 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧 (𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥 𝑦)))
137, 12bitrd 268 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 (𝑥 𝑦)))
14133anbi3d 1554 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦)) ↔ (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))))
153, 14rexeqbidv 3292 . . . . . . 7 (𝑘 = 𝐾 → (∃𝑧 ∈ (Atoms‘𝑘)(𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦)) ↔ ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))))
1615imbi2d 329 . . . . . 6 (𝑘 = 𝐾 → ((𝑥𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧𝑥𝑧𝑦𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ↔ (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)))))
173, 16raleqbidv 3291 . . . . 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 3291 . . . 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 6353 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
20 ishlat.b . . . . . 6 𝐵 = (Base‘𝐾)
2119, 20syl6eqr 2812 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
22 fveq2 6353 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (lt‘𝑘) = (lt‘𝐾))
23 ishlat.s . . . . . . . . . . . 12 < = (lt‘𝐾)
2422, 23syl6eqr 2812 . . . . . . . . . . 11 (𝑘 = 𝐾 → (lt‘𝑘) = < )
2524breqd 4815 . . . . . . . . . 10 (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ (0.‘𝑘) < 𝑥))
26 fveq2 6353 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
27 ishlat.z . . . . . . . . . . . 12 0 = (0.‘𝐾)
2826, 27syl6eqr 2812 . . . . . . . . . . 11 (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
2928breq1d 4814 . . . . . . . . . 10 (𝑘 = 𝐾 → ((0.‘𝑘) < 𝑥0 < 𝑥))
3025, 29bitrd 268 . . . . . . . . 9 (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥0 < 𝑥))
3124breqd 4815 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑥(lt‘𝑘)𝑦𝑥 < 𝑦))
3230, 31anbi12d 749 . . . . . . . 8 (𝑘 = 𝐾 → (((0.‘𝑘)(lt‘𝑘)𝑥𝑥(lt‘𝑘)𝑦) ↔ ( 0 < 𝑥𝑥 < 𝑦)))
3324breqd 4815 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑦(lt‘𝑘)𝑧𝑦 < 𝑧))
3424breqd 4815 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < (1.‘𝑘)))
35 fveq2 6353 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
36 ishlat.u . . . . . . . . . . . 12 1 = (1.‘𝐾)
3735, 36syl6eqr 2812 . . . . . . . . . . 11 (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
3837breq2d 4816 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑧 < (1.‘𝑘) ↔ 𝑧 < 1 ))
3934, 38bitrd 268 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < 1 ))
4033, 39anbi12d 749 . . . . . . . 8 (𝑘 = 𝐾 → ((𝑦(lt‘𝑘)𝑧𝑧(lt‘𝑘)(1.‘𝑘)) ↔ (𝑦 < 𝑧𝑧 < 1 )))
4132, 40anbi12d 749 . . . . . . 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 3292 . . . . . 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 3292 . . . . 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 3292 . . . 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 749 . . 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 35159 . . 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 3507 . 2 (𝐾 ∈ HL ↔ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))
48 elin 3939 . . . . 5 (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
4948anbi1i 733 . . . 4 ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50 elin 3939 . . . 4 (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ↔ (𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat))
51 df-3an 1074 . . . 4 ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
5249, 50, 513bitr4ri 293 . . 3 ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ↔ 𝐾 ∈ ((OML ∩ CLat) ∩ CvLat))
5352anbi1i 733 . 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 267 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  cin 3714   class class class wbr 4804  cfv 6049  (class class class)co 6814  Basecbs 16079  lecple 16170  ltcplt 17162  joincjn 17165  0.cp0 17258  1.cp1 17259  CLatccla 17328  OMLcoml 34983  Atomscatm 35071  CvLatclc 35073  HLchlt 35158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817  df-hlat 35159
This theorem is referenced by:  ishlat2  35161  ishlat3N  35162  hlomcmcv  35164
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