Theorem hlexch1 39369

Description: A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)

Hypotheses

Ref	Expression
hlsuprexch.b	⊢ 𝐵 = (Base‘𝐾)
hlsuprexch.l	⊢ ≤ = (le‘𝐾)
hlsuprexch.j	⊢ ∨ = (join‘𝐾)
hlsuprexch.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
hlexch1	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Proof of Theorem hlexch1

Step	Hyp	Ref	Expression
1		hlcvl 39345	. 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
2		hlsuprexch.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
3		hlsuprexch.l	. . 3 ⊢ ≤ = (le‘𝐾)
4		hlsuprexch.j	. . 3 ⊢ ∨ = (join‘𝐾)
5		hlsuprexch.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6	2, 3, 4, 5	cvlexch1 39314	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
7	1, 6	syl3an1 1163	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  lecple 17203  joincjn 18252  Atomscatm 39249  CvLatclc 39251  HLchlt 39336

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-ov 7372  df-cvlat 39308  df-hlat 39337

This theorem is referenced by:  cvratlem  39408  4noncolr3  39440  3dimlem4a  39450  3dimlem4OLDN  39452  ps-2  39465  4atlem0a  39580