Theorem hlexch1 40134

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 40111	. 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 40080	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
7	1, 6	syl3an1 1181	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  lecple 17318  joincjn 18368  Atomscatm 40015  CvLatclc 40017  HLchlt 40102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-cvlat 40074  df-hlat 40103

This theorem is referenced by:  cvratlem  40173  4noncolr3  40205  3dimlem4a  40215  3dimlem4OLDN  40217  ps-2  40230  4atlem0a  40345