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Definition df-hlat 38269
Description: Define the class of Hilbert lattices, which are complete, atomic lattices satisfying the superposition principle and minimum height. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
df-hlat HL = {𝑙 ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (βˆ€π‘Ž ∈ (Atomsβ€˜π‘™)βˆ€π‘ ∈ (Atomsβ€˜π‘™)(π‘Ž β‰  𝑏 β†’ βˆƒπ‘ ∈ (Atomsβ€˜π‘™)(𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏))) ∧ βˆƒπ‘Ž ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)(((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏) ∧ (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™))))}
Distinct variable group:   𝑐,𝑙,π‘Ž,𝑏

Detailed syntax breakdown of Definition df-hlat
StepHypRef Expression
1 chlt 38268 . 2 class HL
2 va . . . . . . . . 9 setvar π‘Ž
32cv 1541 . . . . . . . 8 class π‘Ž
4 vb . . . . . . . . 9 setvar 𝑏
54cv 1541 . . . . . . . 8 class 𝑏
63, 5wne 2941 . . . . . . 7 wff π‘Ž β‰  𝑏
7 vc . . . . . . . . . . 11 setvar 𝑐
87cv 1541 . . . . . . . . . 10 class 𝑐
98, 3wne 2941 . . . . . . . . 9 wff 𝑐 β‰  π‘Ž
108, 5wne 2941 . . . . . . . . 9 wff 𝑐 β‰  𝑏
11 vl . . . . . . . . . . . . 13 setvar 𝑙
1211cv 1541 . . . . . . . . . . . 12 class 𝑙
13 cjn 18264 . . . . . . . . . . . 12 class join
1412, 13cfv 6544 . . . . . . . . . . 11 class (joinβ€˜π‘™)
153, 5, 14co 7409 . . . . . . . . . 10 class (π‘Ž(joinβ€˜π‘™)𝑏)
16 cple 17204 . . . . . . . . . . 11 class le
1712, 16cfv 6544 . . . . . . . . . 10 class (leβ€˜π‘™)
188, 15, 17wbr 5149 . . . . . . . . 9 wff 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏)
199, 10, 18w3a 1088 . . . . . . . 8 wff (𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏))
20 catm 38181 . . . . . . . . 9 class Atoms
2112, 20cfv 6544 . . . . . . . 8 class (Atomsβ€˜π‘™)
2219, 7, 21wrex 3071 . . . . . . 7 wff βˆƒπ‘ ∈ (Atomsβ€˜π‘™)(𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏))
236, 22wi 4 . . . . . 6 wff (π‘Ž β‰  𝑏 β†’ βˆƒπ‘ ∈ (Atomsβ€˜π‘™)(𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏)))
2423, 4, 21wral 3062 . . . . 5 wff βˆ€π‘ ∈ (Atomsβ€˜π‘™)(π‘Ž β‰  𝑏 β†’ βˆƒπ‘ ∈ (Atomsβ€˜π‘™)(𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏)))
2524, 2, 21wral 3062 . . . 4 wff βˆ€π‘Ž ∈ (Atomsβ€˜π‘™)βˆ€π‘ ∈ (Atomsβ€˜π‘™)(π‘Ž β‰  𝑏 β†’ βˆƒπ‘ ∈ (Atomsβ€˜π‘™)(𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏)))
26 cp0 18376 . . . . . . . . . . 11 class 0.
2712, 26cfv 6544 . . . . . . . . . 10 class (0.β€˜π‘™)
28 cplt 18261 . . . . . . . . . . 11 class lt
2912, 28cfv 6544 . . . . . . . . . 10 class (ltβ€˜π‘™)
3027, 3, 29wbr 5149 . . . . . . . . 9 wff (0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž
313, 5, 29wbr 5149 . . . . . . . . 9 wff π‘Ž(ltβ€˜π‘™)𝑏
3230, 31wa 397 . . . . . . . 8 wff ((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏)
335, 8, 29wbr 5149 . . . . . . . . 9 wff 𝑏(ltβ€˜π‘™)𝑐
34 cp1 18377 . . . . . . . . . . 11 class 1.
3512, 34cfv 6544 . . . . . . . . . 10 class (1.β€˜π‘™)
368, 35, 29wbr 5149 . . . . . . . . 9 wff 𝑐(ltβ€˜π‘™)(1.β€˜π‘™)
3733, 36wa 397 . . . . . . . 8 wff (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™))
3832, 37wa 397 . . . . . . 7 wff (((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏) ∧ (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™)))
39 cbs 17144 . . . . . . . 8 class Base
4012, 39cfv 6544 . . . . . . 7 class (Baseβ€˜π‘™)
4138, 7, 40wrex 3071 . . . . . 6 wff βˆƒπ‘ ∈ (Baseβ€˜π‘™)(((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏) ∧ (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™)))
4241, 4, 40wrex 3071 . . . . 5 wff βˆƒπ‘ ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)(((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏) ∧ (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™)))
4342, 2, 40wrex 3071 . . . 4 wff βˆƒπ‘Ž ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)(((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏) ∧ (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™)))
4425, 43wa 397 . . 3 wff (βˆ€π‘Ž ∈ (Atomsβ€˜π‘™)βˆ€π‘ ∈ (Atomsβ€˜π‘™)(π‘Ž β‰  𝑏 β†’ βˆƒπ‘ ∈ (Atomsβ€˜π‘™)(𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏))) ∧ βˆƒπ‘Ž ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)(((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏) ∧ (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™))))
45 coml 38093 . . . . 5 class OML
46 ccla 18451 . . . . 5 class CLat
4745, 46cin 3948 . . . 4 class (OML ∩ CLat)
48 clc 38183 . . . 4 class CvLat
4947, 48cin 3948 . . 3 class ((OML ∩ CLat) ∩ CvLat)
5044, 11, 49crab 3433 . 2 class {𝑙 ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (βˆ€π‘Ž ∈ (Atomsβ€˜π‘™)βˆ€π‘ ∈ (Atomsβ€˜π‘™)(π‘Ž β‰  𝑏 β†’ βˆƒπ‘ ∈ (Atomsβ€˜π‘™)(𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏))) ∧ βˆƒπ‘Ž ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)(((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏) ∧ (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™))))}
511, 50wceq 1542 1 wff HL = {𝑙 ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (βˆ€π‘Ž ∈ (Atomsβ€˜π‘™)βˆ€π‘ ∈ (Atomsβ€˜π‘™)(π‘Ž β‰  𝑏 β†’ βˆƒπ‘ ∈ (Atomsβ€˜π‘™)(𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏))) ∧ βˆƒπ‘Ž ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)(((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏) ∧ (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™))))}
Colors of variables: wff setvar class
This definition is referenced by:  ishlat1  38270
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