Theorem ishlat1 39375

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 6881	. . . . . 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 6881	. . . . . . . . . . . 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 5135	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(le‘𝑘)(𝑥(join‘𝑘)𝑦) ↔ 𝑧 ≤ (𝑥(join‘𝑘)𝑦)))
8		fveq2 6881	. . . . . . . . . . . . 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 7427	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑦) = (𝑥 ∨ 𝑦))
12	11	breq2d 5136	. . . . . . . . . 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 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 3330	. . . . . . 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 3329	. . . . 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 3329	. . . 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 6881	. . . . . 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 6881	. . . . . . . . . . . 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 5135	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((0.‘𝑘)(lt‘𝑘)𝑥 ↔ (0.‘𝑘) < 𝑥))
26		fveq2 6881	. . . . . . . . . . . 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 5134	. . . . . . . . . 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 5135	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑥(lt‘𝑘)𝑦 ↔ 𝑥 < 𝑦))
32	30, 31	anbi12d 632	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((0.‘𝑘)(lt‘𝑘)𝑥 ∧ 𝑥(lt‘𝑘)𝑦) ↔ ( 0 < 𝑥 ∧ 𝑥 < 𝑦)))
33	24	breqd 5135	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑦(lt‘𝑘)𝑧 ↔ 𝑦 < 𝑧))
34	24	breqd 5135	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑧(lt‘𝑘)(1.‘𝑘) ↔ 𝑧 < (1.‘𝑘)))
35		fveq2 6881	. . . . . . . . . . . 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 5136	. . . . . . . . . 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 632	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑦(lt‘𝑘)𝑧 ∧ 𝑧(lt‘𝑘)(1.‘𝑘)) ↔ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
41	32, 40	anbi12d 632	. . . . . . 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 3330	. . . . . 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 3330	. . . . 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 3330	. . . 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 632	. . 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 39374	. . 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 3679	. 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 3947	. . . . 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 3947	. . . 4 ⊢ (𝐾 ∈ ((OML ∩ CLat) ∩ CvLat) ↔ (𝐾 ∈ (OML ∩ CLat) ∧ 𝐾 ∈ CvLat))
51		df-3an 1088	. . . 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 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 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 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ∩ cin 3930   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  lecple 17283  ltcplt 18325  joincjn 18328  0.cp0 18438  1.cp1 18439  CLatccla 18513  OMLcoml 39198  Atomscatm 39286  CvLatclc 39288  HLchlt 39373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-hlat 39374

This theorem is referenced by:  ishlat2  39376  ishlat3N  39377  hlomcmcv  39379