Theorem hlsupr 39405

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 2735	. . . 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 39400	. . 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 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  lecple 17278  joincjn 18323  Atomscatm 39281  HLchlt 39368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408  df-cvlat 39340  df-hlat 39369

This theorem is referenced by:  hlsupr2  39406  atbtwnexOLDN  39466  atbtwnex  39467  cdlemb  39813  lhpexle2lem  40028  lhpexle3lem  40030  cdlemf1  40580  cdlemg35  40732