Theorem hlsupr 39756

Description: A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)

Hypotheses

Ref	Expression
hlsupr.l	⊢ ≤ = (le‘𝐾)
hlsupr.j	⊢ ∨ = (join‘𝐾)
hlsupr.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
hlsupr	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))

Distinct variable groups:   𝐴,𝑟   𝐾,𝑟   𝑃,𝑟   𝑄,𝑟

Allowed substitution hints:   ∨ (𝑟)   ≤ (𝑟)

Proof of Theorem hlsupr

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		hlsupr.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		hlsupr.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		hlsupr.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	hlsuprexch 39751	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ≠ 𝑄 → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑟 ∈ (Base‘𝐾)((¬ 𝑃 ≤ 𝑟 ∧ 𝑃 ≤ (𝑟 ∨ 𝑄)) → 𝑄 ≤ (𝑟 ∨ 𝑃))))
6	5	simpld 494	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))))
7	6	imp 406	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  joincjn 18246  Atomscatm 39633  HLchlt 39720

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-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-cvlat 39692  df-hlat 39721

This theorem is referenced by:  hlsupr2  39757  atbtwnexOLDN  39817  atbtwnex  39818  cdlemb  40164  lhpexle2lem  40379  lhpexle3lem  40381  cdlemf1  40931  cdlemg35  41083