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Theorem ishlat1 38222
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 𝐡 = (Baseβ€˜πΎ)
ishlat.l ≀ = (leβ€˜πΎ)
ishlat.s < = (ltβ€˜πΎ)
ishlat.j ∨ = (joinβ€˜πΎ)
ishlat.z 0 = (0.β€˜πΎ)
ishlat.u 1 = (1.β€˜πΎ)
ishlat.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
ishlat1 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐴   π‘₯,𝐡,𝑦,𝑧   π‘₯,𝐾,𝑦,𝑧
Allowed substitution hints:   < (π‘₯,𝑦,𝑧)   1 (π‘₯,𝑦,𝑧)   ∨ (π‘₯,𝑦,𝑧)   ≀ (π‘₯,𝑦,𝑧)   0 (π‘₯,𝑦,𝑧)

Proof of Theorem ishlat1
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6892 . . . . . 6 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
2 ishlat.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
31, 2eqtr4di 2791 . . . . 5 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
4 fveq2 6892 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = (leβ€˜πΎ))
5 ishlat.l . . . . . . . . . . . 12 ≀ = (leβ€˜πΎ)
64, 5eqtr4di 2791 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = ≀ )
76breqd 5160 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦) ↔ 𝑧 ≀ (π‘₯(joinβ€˜π‘˜)𝑦)))
8 fveq2 6892 . . . . . . . . . . . . 13 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = (joinβ€˜πΎ))
9 ishlat.j . . . . . . . . . . . . 13 ∨ = (joinβ€˜πΎ)
108, 9eqtr4di 2791 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = ∨ )
1110oveqd 7426 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (π‘₯(joinβ€˜π‘˜)𝑦) = (π‘₯ ∨ 𝑦))
1211breq2d 5161 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (𝑧 ≀ (π‘₯(joinβ€˜π‘˜)𝑦) ↔ 𝑧 ≀ (π‘₯ ∨ 𝑦)))
137, 12bitrd 279 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦) ↔ 𝑧 ≀ (π‘₯ ∨ 𝑦)))
14133anbi3d 1443 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦)) ↔ (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))))
153, 14rexeqbidv 3344 . . . . . . 7 (π‘˜ = 𝐾 β†’ (βˆƒπ‘§ ∈ (Atomsβ€˜π‘˜)(𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦)) ↔ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))))
1615imbi2d 341 . . . . . 6 (π‘˜ = 𝐾 β†’ ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ (Atomsβ€˜π‘˜)(𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦))) ↔ (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦)))))
173, 16raleqbidv 3343 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆ€π‘¦ ∈ (Atomsβ€˜π‘˜)(π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ (Atomsβ€˜π‘˜)(𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦))) ↔ βˆ€π‘¦ ∈ 𝐴 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦)))))
183, 17raleqbidv 3343 . . . 4 (π‘˜ = 𝐾 β†’ (βˆ€π‘₯ ∈ (Atomsβ€˜π‘˜)βˆ€π‘¦ ∈ (Atomsβ€˜π‘˜)(π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ (Atomsβ€˜π‘˜)(𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦))) ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦)))))
19 fveq2 6892 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
20 ishlat.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
2119, 20eqtr4di 2791 . . . . 5 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
22 fveq2 6892 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (ltβ€˜π‘˜) = (ltβ€˜πΎ))
23 ishlat.s . . . . . . . . . . . 12 < = (ltβ€˜πΎ)
2422, 23eqtr4di 2791 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (ltβ€˜π‘˜) = < )
2524breqd 5160 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ ((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ↔ (0.β€˜π‘˜) < π‘₯))
26 fveq2 6892 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (0.β€˜π‘˜) = (0.β€˜πΎ))
27 ishlat.z . . . . . . . . . . . 12 0 = (0.β€˜πΎ)
2826, 27eqtr4di 2791 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (0.β€˜π‘˜) = 0 )
2928breq1d 5159 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ ((0.β€˜π‘˜) < π‘₯ ↔ 0 < π‘₯))
3025, 29bitrd 279 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ ((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ↔ 0 < π‘₯))
3124breqd 5160 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (π‘₯(ltβ€˜π‘˜)𝑦 ↔ π‘₯ < 𝑦))
3230, 31anbi12d 632 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ∧ π‘₯(ltβ€˜π‘˜)𝑦) ↔ ( 0 < π‘₯ ∧ π‘₯ < 𝑦)))
3324breqd 5160 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (𝑦(ltβ€˜π‘˜)𝑧 ↔ 𝑦 < 𝑧))
3424breqd 5160 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜) ↔ 𝑧 < (1.β€˜π‘˜)))
35 fveq2 6892 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (1.β€˜π‘˜) = (1.β€˜πΎ))
36 ishlat.u . . . . . . . . . . . 12 1 = (1.β€˜πΎ)
3735, 36eqtr4di 2791 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (1.β€˜π‘˜) = 1 )
3837breq2d 5161 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (𝑧 < (1.β€˜π‘˜) ↔ 𝑧 < 1 ))
3934, 38bitrd 279 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜) ↔ 𝑧 < 1 ))
4033, 39anbi12d 632 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((𝑦(ltβ€˜π‘˜)𝑧 ∧ 𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜)) ↔ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
4132, 40anbi12d 632 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ∧ π‘₯(ltβ€˜π‘˜)𝑦) ∧ (𝑦(ltβ€˜π‘˜)𝑧 ∧ 𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜))) ↔ (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))
4221, 41rexeqbidv 3344 . . . . . 6 (π‘˜ = 𝐾 β†’ (βˆƒπ‘§ ∈ (Baseβ€˜π‘˜)(((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ∧ π‘₯(ltβ€˜π‘˜)𝑦) ∧ (𝑦(ltβ€˜π‘˜)𝑧 ∧ 𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜))) ↔ βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))
4321, 42rexeqbidv 3344 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆƒπ‘¦ ∈ (Baseβ€˜π‘˜)βˆƒπ‘§ ∈ (Baseβ€˜π‘˜)(((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ∧ π‘₯(ltβ€˜π‘˜)𝑦) ∧ (𝑦(ltβ€˜π‘˜)𝑧 ∧ 𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜))) ↔ βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))
4421, 43rexeqbidv 3344 . . . 4 (π‘˜ = 𝐾 β†’ (βˆƒπ‘₯ ∈ (Baseβ€˜π‘˜)βˆƒπ‘¦ ∈ (Baseβ€˜π‘˜)βˆƒπ‘§ ∈ (Baseβ€˜π‘˜)(((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ∧ π‘₯(ltβ€˜π‘˜)𝑦) ∧ (𝑦(ltβ€˜π‘˜)𝑧 ∧ 𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜))) ↔ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))
4518, 44anbi12d 632 . . 3 (π‘˜ = 𝐾 β†’ ((βˆ€π‘₯ ∈ (Atomsβ€˜π‘˜)βˆ€π‘¦ ∈ (Atomsβ€˜π‘˜)(π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ (Atomsβ€˜π‘˜)(𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦))) ∧ βˆƒπ‘₯ ∈ (Baseβ€˜π‘˜)βˆƒπ‘¦ ∈ (Baseβ€˜π‘˜)βˆƒπ‘§ ∈ (Baseβ€˜π‘˜)(((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ∧ π‘₯(ltβ€˜π‘˜)𝑦) ∧ (𝑦(ltβ€˜π‘˜)𝑧 ∧ 𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜)))) ↔ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
46 df-hlat 38221 . . 3 HL = {π‘˜ ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (βˆ€π‘₯ ∈ (Atomsβ€˜π‘˜)βˆ€π‘¦ ∈ (Atomsβ€˜π‘˜)(π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ (Atomsβ€˜π‘˜)(𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦))) ∧ βˆƒπ‘₯ ∈ (Baseβ€˜π‘˜)βˆƒπ‘¦ ∈ (Baseβ€˜π‘˜)βˆƒπ‘§ ∈ (Baseβ€˜π‘˜)(((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ∧ π‘₯(ltβ€˜π‘˜)𝑦) ∧ (𝑦(ltβ€˜π‘˜)𝑧 ∧ 𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜))))}
4745, 46elrab2 3687 . 2 (𝐾 ∈ HL ↔ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
48 elin 3965 . . . . 5 (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
4948anbi1i 625 . . . 4 ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50 elin 3965 . . . 4 (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ↔ (𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat))
51 df-3an 1090 . . . 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 625 . 2 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))) ↔ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
5447, 53bitr4i 278 1 (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3948   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  ltcplt 18261  joincjn 18264  0.cp0 18376  1.cp1 18377  CLatccla 18451  OMLcoml 38045  Atomscatm 38133  CvLatclc 38135  HLchlt 38220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-hlat 38221
This theorem is referenced by:  ishlat2  38223  ishlat3N  38224  hlomcmcv  38226
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