Theorem ishlatiN 39991

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 1356	. 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 475	. 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 39989	. 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 723	1 ⊢ 𝐾 ∈ HL

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∃wrex 3089   class class class wbr 5105  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  lecple 17307  ltcplt 18354  joincjn 18357  0.cp0 18467  1.cp1 18468  CLatccla 18544  OMLcoml 39811  Atomscatm 39899  AtLatcal 39900  HLchlt 39986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403  df-cvlat 39958  df-hlat 39987

This theorem is referenced by: (None)