Theorem ishlat1 37052

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 6695	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2		ishlat.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	eqtr4di 2789	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4		fveq2 6695	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5		ishlat.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
7	6	breqd 5050	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥(join‘𝑘)𝑦)))
8		fveq2 6695	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9		ishlat.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
10	8, 9	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
11	10	oveqd 7208	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑦) = (𝑥 ∨ 𝑦))
12	11	breq2d 5051	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧 ≤ (𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥 ∨ 𝑦)))
13	7, 12	bitrd 282	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥 ∨ 𝑦)))
14	13	3anbi3d 1444	. . . . . . . 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 3304	. . . . . . 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 344	. . . . . 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 3303	. . . . 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 3303	. . . 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 6695	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
20		ishlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
21	19, 20	eqtr4di 2789	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
22		fveq2 6695	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (lt‘𝑘) = (lt‘𝐾))
23		ishlat.s	. . . . . . . . . . . 12 ⊢ < = (lt‘𝐾)
24	22, 23	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (lt‘𝑘) = < )
25	24	breqd 5050	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ (0.‘𝑘) < 𝑥))
26		fveq2 6695	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
27		ishlat.z	. . . . . . . . . . . 12 ⊢ 0 = (0.‘𝐾)
28	26, 27	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
29	28	breq1d 5049	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘) < 𝑥 ↔ 0 < 𝑥))
30	25, 29	bitrd 282	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ 0 < 𝑥))
31	24	breqd 5050	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥(lt‘𝑘)𝑦 ↔ 𝑥 < 𝑦))
32	30, 31	anbi12d 634	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ↔ ( 0 < 𝑥 ∧ 𝑥 < 𝑦)))
33	24	breqd 5050	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑦(lt‘𝑘)𝑧 ↔ 𝑦 < 𝑧))
34	24	breqd 5050	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < (1.‘𝑘)))
35		fveq2 6695	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
36		ishlat.u	. . . . . . . . . . . 12 ⊢ 1 = (1.‘𝐾)
37	35, 36	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
38	37	breq2d 5051	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧 < (1.‘𝑘) ↔ 𝑧 < 1 ))
39	34, 38	bitrd 282	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < 1 ))
40	33, 39	anbi12d 634	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘)) ↔ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
41	32, 40	anbi12d 634	. . . . . . 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 3304	. . . . . 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 3304	. . . . 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 3304	. . . 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 634	. . 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 37051	. . 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 3594	. 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 3869	. . . . 5 ⊢ (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
49	48	anbi1i 627	. . . 4 ⊢ ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50		elin 3869	. . . 4 ⊢ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ↔ (𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat))
51		df-3an 1091	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
52	49, 50, 51	3bitr4ri 307	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ↔ 𝐾 ∈ ((OML ∩ CLat) ∩ CvLat))
53	52	anbi1i 627	. 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 281	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 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  ∀wral 3051  ∃wrex 3052   ∩ cin 3852   class class class wbr 5039  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  lecple 16756  ltcplt 17769  joincjn 17772  0.cp0 17883  1.cp1 17884  CLatccla 17958  OMLcoml 36875  Atomscatm 36963  CvLatclc 36965  HLchlt 37050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194  df-hlat 37051

This theorem is referenced by:  ishlat2  37053  ishlat3N  37054  hlomcmcv  37056