Theorem ishlat1 37814

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 6842	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2		ishlat.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	eqtr4di 2794	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5		ishlat.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
7	6	breqd 5116	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥(join‘𝑘)𝑦)))
8		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9		ishlat.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
10	8, 9	eqtr4di 2794	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
11	10	oveqd 7374	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑦) = (𝑥 ∨ 𝑦))
12	11	breq2d 5117	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧 ≤ (𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥 ∨ 𝑦)))
13	7, 12	bitrd 278	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥 ∨ 𝑦)))
14	13	3anbi3d 1442	. . . . . . . 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 3320	. . . . . . 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 3319	. . . . 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 3319	. . . 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 6842	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
20		ishlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
21	19, 20	eqtr4di 2794	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
22		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (lt‘𝑘) = (lt‘𝐾))
23		ishlat.s	. . . . . . . . . . . 12 ⊢ < = (lt‘𝐾)
24	22, 23	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (lt‘𝑘) = < )
25	24	breqd 5116	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ (0.‘𝑘) < 𝑥))
26		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
27		ishlat.z	. . . . . . . . . . . 12 ⊢ 0 = (0.‘𝐾)
28	26, 27	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
29	28	breq1d 5115	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘) < 𝑥 ↔ 0 < 𝑥))
30	25, 29	bitrd 278	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ 0 < 𝑥))
31	24	breqd 5116	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥(lt‘𝑘)𝑦 ↔ 𝑥 < 𝑦))
32	30, 31	anbi12d 631	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ↔ ( 0 < 𝑥 ∧ 𝑥 < 𝑦)))
33	24	breqd 5116	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑦(lt‘𝑘)𝑧 ↔ 𝑦 < 𝑧))
34	24	breqd 5116	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < (1.‘𝑘)))
35		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
36		ishlat.u	. . . . . . . . . . . 12 ⊢ 1 = (1.‘𝐾)
37	35, 36	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
38	37	breq2d 5117	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧 < (1.‘𝑘) ↔ 𝑧 < 1 ))
39	34, 38	bitrd 278	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < 1 ))
40	33, 39	anbi12d 631	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘)) ↔ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
41	32, 40	anbi12d 631	. . . . . . 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 3320	. . . . . 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 3320	. . . . 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 3320	. . . 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 631	. . 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 37813	. . 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 3648	. 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 3926	. . . . 5 ⊢ (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
49	48	anbi1i 624	. . . 4 ⊢ ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50		elin 3926	. . . 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 303	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ↔ 𝐾 ∈ ((OML ∩ CLat) ∩ CvLat))
53	52	anbi1i 624	. 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 277	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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073   ∩ cin 3909   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  ltcplt 18197  joincjn 18200  0.cp0 18312  1.cp1 18313  CLatccla 18387  OMLcoml 37637  Atomscatm 37725  CvLatclc 37727  HLchlt 37812

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360  df-hlat 37813

This theorem is referenced by:  ishlat2  37815  ishlat3N  37816  hlomcmcv  37818