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Metamath Proof Explorer

Theorem ishlatiN

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

Ref Expression
Hypotheses 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 ishlatiN 
|- K e. HL

Proof

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