Theorem ishlatiN 39815

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 1341	. 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 470	. 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 39813	. 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 712	1 ⊢ 𝐾 ∈ HL

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  lecple 17218  ltcplt 18265  joincjn 18268  0.cp0 18378  1.cp1 18379  CLatccla 18455  OMLcoml 39635  Atomscatm 39723  AtLatcal 39724  HLchlt 39810

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-ov 7363  df-cvlat 39782  df-hlat 39811

This theorem is referenced by: (None)