Theorem omlol 40390

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 2760	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2760	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2760	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2760	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2760	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 40388	. 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 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  lecple 17440  occoc 17441  joincjn 18490  meetcmee 18491  OLcol 40324  OMLcoml 40325

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-oml 40329

This theorem is used by:  omlop  40391  omllat  40392  omllaw3  40395  omllaw4  40396  cmtcomlemN  40398  cmtbr2N  40403  cmtbr3N  40404  omlfh1N  40408  omlfh3N  40409  omlspjN  40411  hlol  40511