Theorem omlol 39258

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 2730	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2730	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2730	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2730	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2730	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 39256	. 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 497	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2110  ∀wral 3045   class class class wbr 5089  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  lecple 17160  occoc 17161  joincjn 18209  meetcmee 18210  OLcol 39192  OMLcoml 39193

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 2112  ax-9 2120  ax-ext 2702

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 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-iota 6433  df-fv 6485  df-ov 7344  df-oml 39197

This theorem is referenced by:  omlop  39259  omllat  39260  omllaw3  39263  omllaw4  39264  cmtcomlemN  39266  cmtbr2N  39271  cmtbr3N  39272  omlfh1N  39276  omlfh3N  39277  omlspjN  39279  hlol  39379