Theorem ishlat1 40555

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 6896	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2		ishlat.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	eqtr4di 2816	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4		fveq2 6896	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5		ishlat.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
7	6	breqd 5117	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥(join‘𝑘)𝑦)))
8		fveq2 6896	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9		ishlat.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
10	8, 9	eqtr4di 2816	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
11	10	oveqd 7447	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑦) = (𝑥 ∨ 𝑦))
12	11	breq2d 5118	. . . . . . . . . 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 1470	. . . . . . . 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 3339	. . . . . . 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 343	. . . . . 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 3338	. . . . 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 3338	. . . 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 6896	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
20		ishlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
21	19, 20	eqtr4di 2816	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
22		fveq2 6896	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (lt‘𝑘) = (lt‘𝐾))
23		ishlat.s	. . . . . . . . . . . 12 ⊢ < = (lt‘𝐾)
24	22, 23	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (lt‘𝑘) = < )
25	24	breqd 5117	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ (0.‘𝑘) < 𝑥))
26		fveq2 6896	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
27		ishlat.z	. . . . . . . . . . . 12 ⊢ 0 = (0.‘𝐾)
28	26, 27	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
29	28	breq1d 5116	. . . . . . . . . 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 5117	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥(lt‘𝑘)𝑦 ↔ 𝑥 < 𝑦))
32	30, 31	anbi12d 644	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ↔ ( 0 < 𝑥 ∧ 𝑥 < 𝑦)))
33	24	breqd 5117	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑦(lt‘𝑘)𝑧 ↔ 𝑦 < 𝑧))
34	24	breqd 5117	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < (1.‘𝑘)))
35		fveq2 6896	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
36		ishlat.u	. . . . . . . . . . . 12 ⊢ 1 = (1.‘𝐾)
37	35, 36	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
38	37	breq2d 5118	. . . . . . . . . 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 644	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘)) ↔ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
41	32, 40	anbi12d 644	. . . . . . 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 3339	. . . . . 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 3339	. . . . 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 3339	. . . 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 644	. . 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 40554	. . 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 3652	. 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 4096	. . . . 5 ⊢ (𝐾 ∈ (OML ∩ CLat) ↔ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat))
49	48	anbi1i 636	. . . 4 ⊢ ((𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat) ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat) ∧ 𝐾 ∈ CvLat))
50		elin 4096	. . . 4 ⊢ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ↔ (𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat))
51		df-3an 1105	. . . 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 636	. 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 )))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ∩ cin 4092   class class class wbr 5106  ‘cfv 6547  (class class class)co 7430  Basecbs 17402  lecple 17450  ltcplt 18497  joincjn 18500  0.cp0 18610  1.cp1 18611  CLatccla 18687  OMLcoml 40378  Atomscatm 40466  CvLatclc 40468  HLchlt 40553

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6501  df-fv 6555  df-ov 7433  df-hlat 40554

This theorem is used by:  ishlat2  40556  ishlat3N  40557  hlomcmcv  40559