Theorem ishlat1 39680

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) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 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.

Step	Hyp	Ref	Expression
1		fveq2 6835	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2		ishlat.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	eqtr4di 2790	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5		ishlat.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
7	6	breqd 5110	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥(join‘𝑘)𝑦)))
8		fveq2 6835	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9		ishlat.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
10	8, 9	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
11	10	oveqd 7377	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑦) = (𝑥 ∨ 𝑦))
12	11	breq2d 5111	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧 ≤ (𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥 ∨ 𝑦)))
13	7, 12	bitrd 279	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥 ∨ 𝑦)))
14	13	3anbi3d 1445	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦)) ↔ (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))))
15	3, 14	rexeqbidv 3318	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∃𝑧 ∈ (Atoms‘𝑘)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦)) ↔ ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))))
16	15	imbi2d 340	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ↔ (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)))))
17	3, 16	raleqbidv 3317	. . . . 5 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (Atoms‘𝑘)(𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)))))
18	3, 17	raleqbidv 3317	. . . 4 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Atoms‘𝑘)∀𝑦 ∈ (Atoms‘𝑘)(𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)))))
19		fveq2 6835	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
20		ishlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
21	19, 20	eqtr4di 2790	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
22		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (lt‘𝑘) = (lt‘𝐾))
23		ishlat.s	. . . . . . . . . . . 12 ⊢ < = (lt‘𝐾)
24	22, 23	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (lt‘𝑘) = < )
25	24	breqd 5110	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ (0.‘𝑘) < 𝑥))
26		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
27		ishlat.z	. . . . . . . . . . . 12 ⊢ 0 = (0.‘𝐾)
28	26, 27	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
29	28	breq1d 5109	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘) < 𝑥 ↔ 0 < 𝑥))
30	25, 29	bitrd 279	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ 0 < 𝑥))
31	24	breqd 5110	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥(lt‘𝑘)𝑦 ↔ 𝑥 < 𝑦))
32	30, 31	anbi12d 633	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ↔ ( 0 < 𝑥 ∧ 𝑥 < 𝑦)))
33	24	breqd 5110	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑦(lt‘𝑘)𝑧 ↔ 𝑦 < 𝑧))
34	24	breqd 5110	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < (1.‘𝑘)))
35		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
36		ishlat.u	. . . . . . . . . . . 12 ⊢ 1 = (1.‘𝐾)
37	35, 36	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
38	37	breq2d 5111	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧 < (1.‘𝑘) ↔ 𝑧 < 1 ))
39	34, 38	bitrd 279	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < 1 ))
40	33, 39	anbi12d 633	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘)) ↔ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
41	32, 40	anbi12d 633	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘))) ↔ (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))
42	21, 41	rexeqbidv 3318	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘))) ↔ ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))
43	21, 42	rexeqbidv 3318	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (Base‘𝑘)∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘))) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))
44	21, 43	rexeqbidv 3318	. . . 4 ⊢ (𝑘 = 𝐾 → (∃𝑥 ∈ (Base‘𝑘)∃𝑦 ∈ (Base‘𝑘)∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘))) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))))
45	18, 44	anbi12d 633	. . 3 ⊢ (𝑘 = 𝐾 → ((∀𝑥 ∈ (Atoms‘𝑘)∀𝑦 ∈ (Atoms‘𝑘)(𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ∧ ∃𝑥 ∈ (Base‘𝑘)∃𝑦 ∈ (Base‘𝑘)∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘)))) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
46		df-hlat 39679	. . 3 ⊢ HL = {𝑘 ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (∀𝑥 ∈ (Atoms‘𝑘)∀𝑦 ∈ (Atoms‘𝑘)(𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝑘)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦))) ∧ ∃𝑥 ∈ (Base‘𝑘)∃𝑦 ∈ (Base‘𝑘)∃𝑧 ∈ (Base‘𝑘)(((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ∧ (𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘))))}
47	45, 46	elrab2 3650	. 2 ⊢ (𝐾 ∈ HL ↔ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
48		elin 3918	. . . . 5 ⊢ (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
49	48	anbi1i 625	. . . 4 ⊢ ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50		elin 3918	. . . 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))
52	49, 50, 51	3bitr4ri 304	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ↔ 𝐾 ∈ ((OML ∩ CLat) ∩ CvLat))
53	52	anbi1i 625	. 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))) ↔ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
54	47, 53	bitr4i 278	1 ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ∩ cin 3901   class class class wbr 5099  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  lecple 17188  ltcplt 18235  joincjn 18238  0.cp0 18348  1.cp1 18349  CLatccla 18425  OMLcoml 39503  Atomscatm 39591  CvLatclc 39593  HLchlt 39678

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-ov 7363  df-hlat 39679

This theorem is referenced by:  ishlat2  39681  ishlat3N  39682  hlomcmcv  39684