Theorem isomliN 34526

Description: Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isomli.0	|- K e. OL
isomli.b	|- B = ( Base ` K )
isomli.l	|- .<_ = ( le ` K )
isomli.j	|- .\/ = ( join ` K )
isomli.m	|- ./\ = ( meet ` K )
isomli.o	|- ._|_ = ( oc ` K )
isomli.7	|- ( ( x e. B /\ y e. B ) -> ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) )

Assertion

Ref	Expression
isomliN	|- K e. OML

Distinct variable groups:   x, y, B   x, K, y

Allowed substitution hints:   .\/ ( x, y)   .<_ ( x, y)   ./\ ( x, y)   ._|_ ( x, y)

Proof of Theorem isomliN

Step	Hyp	Ref	Expression
1		isomli.0	. 2 |- K e. OL
2		isomli.7	. . 3 |- ( ( x e. B /\ y e. B ) -> ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) )
3	2	rgen2a 2977	. 2 |- A. x e. B A. y e. B ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) )
4		isomli.b	. . 3 |- B = ( Base ` K )
5		isomli.l	. . 3 |- .<_ = ( le ` K )
6		isomli.j	. . 3 |- .\/ = ( join ` K )
7		isomli.m	. . 3 |- ./\ = ( meet ` K )
8		isomli.o	. . 3 |- ._|_ = ( oc ` K )
9	4, 5, 6, 7, 8	isoml 34525	. 2 |- ( K e. OML <-> ( K e. OL /\ A. x e. B A. y e. B ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) )
10	1, 3, 9	mpbir2an 955	1 |- K e. OML

Syntax hints:   -> wi 4   /\ wa 384   = 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: (None)