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Theorem ishlat1 37860
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 6843 . . . . . 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 6843 . . . . . . . . . . . 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 5117 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (𝑧(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑦) ↔ 𝑧 ≀ (π‘₯(joinβ€˜π‘˜)𝑦)))
8 fveq2 6843 . . . . . . . . . . . . 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 7375 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (π‘₯(joinβ€˜π‘˜)𝑦) = (π‘₯ ∨ 𝑦))
1211breq2d 5118 . . . . . . . . . 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 3319 . . . . . . 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 3318 . . . . 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 3318 . . . 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 6843 . . . . . 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 6843 . . . . . . . . . . . 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 5117 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ ((0.β€˜π‘˜)(ltβ€˜π‘˜)π‘₯ ↔ (0.β€˜π‘˜) < π‘₯))
26 fveq2 6843 . . . . . . . . . . . 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 5116 . . . . . . . . . 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 5117 . . . . . . . . 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 5117 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (𝑦(ltβ€˜π‘˜)𝑧 ↔ 𝑦 < 𝑧))
3424breqd 5117 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (𝑧(ltβ€˜π‘˜)(1.β€˜π‘˜) ↔ 𝑧 < (1.β€˜π‘˜)))
35 fveq2 6843 . . . . . . . . . . . 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 5118 . . . . . . . . . 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 3319 . . . . . 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 3319 . . . . 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 3319 . . . 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 37859 . . 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 3649 . 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 3927 . . . . 5 (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
4948anbi1i 625 . . . 4 ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50 elin 3927 . . . 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 2940  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3910   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  ltcplt 18202  joincjn 18205  0.cp0 18317  1.cp1 18318  CLatccla 18392  OMLcoml 37683  Atomscatm 37771  CvLatclc 37773  HLchlt 37858
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 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-hlat 37859
This theorem is referenced by:  ishlat2  37861  ishlat3N  37862  hlomcmcv  37864
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