Theorem hlsupr 36524

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 2823	. . . 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 36519	. . 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 497	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))))
7	6	imp 409	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 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  lecple 16574  joincjn 17556  Atomscatm 36401  HLchlt 36488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-cvlat 36460  df-hlat 36489

This theorem is referenced by:  hlsupr2  36525  atbtwnexOLDN  36585  atbtwnex  36586  cdlemb  36932  lhpexle2lem  37147  lhpexle3lem  37149  cdlemf1  37699  cdlemg35  37851