Theorem hlhgt4 40425

Description: A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)

Hypotheses

Ref	Expression
hlhgt4.b	⊢ 𝐵 = (Base‘𝐾)
hlhgt4.s	⊢ < = (lt‘𝐾)
hlhgt4.z	⊢ 0 = (0.‘𝐾)
hlhgt4.u	⊢ 1 = (1.‘𝐾)

Assertion

Ref	Expression
hlhgt4	⊢ (𝐾 ∈ HL → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐾,𝑦,𝑧

Allowed substitution hints:   < (𝑥, 𝑦, 𝑧)   1 (𝑥, 𝑦, 𝑧)   0 (𝑥, 𝑦, 𝑧)

Proof of Theorem hlhgt4

Step	Hyp	Ref	Expression
1		hlhgt4.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2761	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		hlhgt4.s	. . 3 ⊢ < = (lt‘𝐾)
4		eqid 2761	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
5		hlhgt4.z	. . 3 ⊢ 0 = (0.‘𝐾)
6		hlhgt4.u	. . 3 ⊢ 1 = (1.‘𝐾)
7		eqid 2761	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat2 40390	. 2 ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ (Atoms‘𝐾)∀𝑦 ∈ (Atoms‘𝐾)((𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝐾)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝐾)(𝑥(join‘𝐾)𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥(le‘𝐾)𝑧 ∧ 𝑥(le‘𝐾)(𝑧(join‘𝐾)𝑦)) → 𝑦(le‘𝐾)(𝑧(join‘𝐾)𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
9		simprr 785	. 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ (Atoms‘𝐾)∀𝑦 ∈ (Atoms‘𝐾)((𝑥 ≠ 𝑦 → ∃𝑧 ∈ (Atoms‘𝐾)(𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧(le‘𝐾)(𝑥(join‘𝐾)𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥(le‘𝐾)𝑧 ∧ 𝑥(le‘𝐾)(𝑧(join‘𝐾)𝑦)) → 𝑦(le‘𝐾)(𝑧(join‘𝐾)𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
10	8, 9	sylbi 220	1 ⊢ (𝐾 ∈ HL → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (( 0 < 𝑥 ∧ 𝑥 < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))

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 6537  (class class class)co 7418  Basecbs 17380  lecple 17428  ltcplt 18475  joincjn 18478  0.cp0 18588  1.cp1 18589  CLatccla 18665  OMLcoml 40212  Atomscatm 40300  AtLatcal 40301  HLchlt 40387

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 6493  df-fv 6545  df-ov 7421  df-cvlat 40359  df-hlat 40388

This theorem is used by:  hlhgt2  40426  athgt  40493