Theorem omlol 34527

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

Assertion

Ref	Expression
omlol	|- ( K e. OML -> K e. OL )

Proof of Theorem omlol

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` K ) = ( Base ` K )
2		eqid 2622	. . 3 |- ( le ` K ) = ( le ` K )
3		eqid 2622	. . 3 |- ( join ` K ) = ( join ` K )
4		eqid 2622	. . 3 |- ( meet ` K ) = ( meet ` K )
5		eqid 2622	. . 3 |- ( oc ` K ) = ( oc ` K )
6	1, 2, 3, 4, 5	isoml 34525	. 2 |- ( K e. OML <-> ( K e. OL /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) ( x ( le ` K ) y -> y = ( x ( join ` K ) ( y ( meet ` K ) ( ( oc ` K ) ` x ) ) ) ) ) )
7	6	simplbi 476	1 |- ( K e. OML -> K e. OL )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   occoc 15949   joincjn 16944   meetcmee 16945   OLcol 34461   OMLcoml 34462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-oml 34466

This theorem is referenced by:  omlop  34528  omllat  34529  omllaw3  34532  omllaw4  34533  cmtcomlemN  34535  cmtbr2N  34540  cmtbr3N  34541  omlfh1N  34545  omlfh3N  34546  omlspjN  34548  hlol  34648