Theorem ishlatiN 34642

Description: Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ishlati.1	|- K e. OML
ishlati.2	|- K e. CLat
ishlati.3	|- K e. AtLat
ishlati.b	|- B = ( Base ` K )
ishlati.l	|- .<_ = ( le ` K )
ishlati.s	|- .< = ( lt ` K )
ishlati.j	|- .\/ = ( join ` K )
ishlati.z	|- .0. = ( 0. ` K )
ishlati.u	|- .1. = ( 1. ` K )
ishlati.a	|- A = ( Atoms ` K )
ishlati.9	|- A. x e. A A. y e. A ( ( x =/= y -> E. z e. A ( z =/= x /\ z =/= y /\ z .<_ ( x .\/ y ) ) ) /\ A. z e. B ( ( -. x .<_ z /\ x .<_ ( z .\/ y ) ) -> y .<_ ( z .\/ x ) ) )
ishlati.10	|- E. x e. B E. y e. B E. z e. B ( ( .0. .< x /\ x .< y ) /\ ( y .< z /\ z .< .1. ) )

Assertion

Ref	Expression
ishlatiN	|- K e. HL

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

Allowed substitution hints:   .< ( x, y, z)   .1. ( x, y, z)   .\/ ( x, y, z)   .<_ ( x, y, z)   .0. ( x, y, z)

Proof of Theorem ishlatiN

Step	Hyp	Ref	Expression
1		ishlati.1	. . 3 |- K e. OML
2		ishlati.2	. . 3 |- K e. CLat
3		ishlati.3	. . 3 |- K e. AtLat
4	1, 2, 3	3pm3.2i 1239	. 2 |- ( K e. OML /\ K e. CLat /\ K e. AtLat )
5		ishlati.9	. . 3 |- A. x e. A A. y e. A ( ( x =/= y -> E. z e. A ( z =/= x /\ z =/= y /\ z .<_ ( x .\/ y ) ) ) /\ A. z e. B ( ( -. x .<_ z /\ x .<_ ( z .\/ y ) ) -> y .<_ ( z .\/ x ) ) )
6		ishlati.10	. . 3 |- E. x e. B E. y e. B E. z e. B ( ( .0. .< x /\ x .< y ) /\ ( y .< z /\ z .< .1. ) )
7	5, 6	pm3.2i 471	. 2 |- ( A. x e. A A. y e. A ( ( x =/= y -> E. z e. A ( z =/= x /\ z =/= y /\ z .<_ ( x .\/ y ) ) ) /\ A. z e. B ( ( -. x .<_ z /\ x .<_ ( z .\/ y ) ) -> y .<_ ( z .\/ x ) ) ) /\ E. x e. B E. y e. B E. z e. B ( ( .0. .< x /\ x .< y ) /\ ( y .< z /\ z .< .1. ) ) )
8		ishlati.b	. . 3 |- B = ( Base ` K )
9		ishlati.l	. . 3 |- .<_ = ( le ` K )
10		ishlati.s	. . 3 |- .< = ( lt ` K )
11		ishlati.j	. . 3 |- .\/ = ( join ` K )
12		ishlati.z	. . 3 |- .0. = ( 0. ` K )
13		ishlati.u	. . 3 |- .1. = ( 1. ` K )
14		ishlati.a	. . 3 |- A = ( Atoms ` K )
15	8, 9, 10, 11, 12, 13, 14	ishlat2 34640	. 2 |- ( K e. HL <-> ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ ( A. x e. A A. y e. A ( ( x =/= y -> E. z e. A ( z =/= x /\ z =/= y /\ z .<_ ( x .\/ y ) ) ) /\ A. z e. B ( ( -. x .<_ z /\ x .<_ ( z .\/ y ) ) -> y .<_ ( z .\/ x ) ) ) /\ E. x e. B E. y e. B E. z e. B ( ( .0. .< x /\ x .< y ) /\ ( y .< z /\ z .< .1. ) ) ) ) )
16	4, 7, 15	mpbir2an 955	1 |- K e. HL

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   ltcplt 16941   joincjn 16944   0.cp0 17037   1.cp1 17038   CLatccla 17107   OMLcoml 34462   Atomscatm 34550   AtLatcal 34551   HLchlt 34637

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-cvlat 34609  df-hlat 34638

This theorem is referenced by: (None)