Theorem omlol 40121

Description: An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)

Assertion

Ref	Expression
omlol	⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)

Proof of Theorem omlol

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2762	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2762	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2762	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2762	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2762	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 40119	. 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 → 𝑦 = (𝑥(join‘𝐾)(𝑦(meet‘𝐾)((oc‘𝐾)‘𝑥))))))
7	6	simplbi 502	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3078   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  lecple 17355  occoc 17356  joincjn 18405  meetcmee 18406  OLcol 40055  OMLcoml 40056

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420  df-oml 40060

This theorem is used by:  omlop  40122  omllat  40123  omllaw3  40126  omllaw4  40127  cmtcomlemN  40129  cmtbr2N  40134  cmtbr3N  40135  omlfh1N  40139  omlfh3N  40140  omlspjN  40142  hlol  40242