Theorem omlol 36408

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 2821	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2821	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2821	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2821	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2821	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 36406	. 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 500	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3138   class class class wbr 5052  ‘cfv 6341  (class class class)co 7142  Basecbs 16466  lecple 16555  occoc 16556  joincjn 17537  meetcmee 17538  OLcol 36342  OMLcoml 36343

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3488  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-iota 6300  df-fv 6349  df-ov 7145  df-oml 36347

This theorem is referenced by:  omlop  36409  omllat  36410  omllaw3  36413  omllaw4  36414  cmtcomlemN  36416  cmtbr2N  36421  cmtbr3N  36422  omlfh1N  36426  omlfh3N  36427  omlspjN  36429  hlol  36529