Theorem hlexch1 39480

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 39457	. 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 39426	. 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 1541   ∈ wcel 2111   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  lecple 17168  joincjn 18217  Atomscatm 39361  CvLatclc 39363  HLchlt 39448

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-cvlat 39420  df-hlat 39449

This theorem is referenced by:  cvratlem  39519  4noncolr3  39551  3dimlem4a  39561  3dimlem4OLDN  39563  ps-2  39576  4atlem0a  39691