Theorem ishlatiN 40412

Description: Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ishlati.1	⊢ 𝐾 ∈ OML
ishlati.2	⊢ 𝐾 ∈ CLat
ishlati.3	⊢ 𝐾 ∈ AtLat
ishlati.b	⊢ 𝐵 = (Base‘𝐾)
ishlati.l	⊢ ≤ = (le‘𝐾)
ishlati.s	⊢ < = (lt‘𝐾)
ishlati.j	⊢ ∨ = (join‘𝐾)
ishlati.z	⊢ 0 = (0.‘𝐾)
ishlati.u	⊢ 1 = (1.‘𝐾)
ishlati.a	⊢ 𝐴 = (Atoms‘𝐾)
ishlati.9	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥)))
ishlati.10	⊢ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))

Assertion

Ref	Expression
ishlatiN	⊢ 𝐾 ∈ HL

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 ishlatiN

Step	Hyp	Ref	Expression
1		ishlati.1	. . 3 ⊢ 𝐾 ∈ OML
2		ishlati.2	. . 3 ⊢ 𝐾 ∈ CLat
3		ishlati.3	. . 3 ⊢ 𝐾 ∈ AtLat
4	1, 2, 3	3pm3.2i 1358	. 2 ⊢ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)
5		ishlati.9	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥)))
6		ishlati.10	. . 3 ⊢ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))
7	5, 6	pm3.2i 476	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
8		ishlati.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
9		ishlati.l	. . 3 ⊢ ≤ = (le‘𝐾)
10		ishlati.s	. . 3 ⊢ < = (lt‘𝐾)
11		ishlati.j	. . 3 ⊢ ∨ = (join‘𝐾)
12		ishlati.z	. . 3 ⊢ 0 = (0.‘𝐾)
13		ishlati.u	. . 3 ⊢ 1 = (1.‘𝐾)
14		ishlati.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
15	8, 9, 10, 11, 12, 13, 14	ishlat2 40410	. 2 ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
16	4, 7, 15	mpbir2an 724	1 ⊢ 𝐾 ∈ HL

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  lecple 17435  ltcplt 18482  joincjn 18485  0.cp0 18595  1.cp1 18596  CLatccla 18672  OMLcoml 40232  Atomscatm 40320  AtLatcal 40321  HLchlt 40407

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-cvlat 40379  df-hlat 40408

This theorem is used by: (None)