Theorem hlsupr 34672

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 2622	. . . 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 34667	. . 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 475	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))))
7	6	imp 445	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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  lecple 15948  joincjn 16944  Atomscatm 34550  HLchlt 34637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  hlsupr2  34673  atbtwnexOLDN  34733  atbtwnex  34734  cdlemb  35080  lhpexle2lem  35295  lhpexle3lem  35297  cdlemf1  35849  cdlemg35  36001