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Theorem ishlatiN 30215
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
StepHypRef Expression
1 ishlati.1 . . 3  |-  K  e. 
OML
2 ishlati.2 . . 3  |-  K  e. 
CLat
3 ishlati.3 . . 3  |-  K  e. 
AtLat
41, 2, 33pm3.2i 1133 . 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.  ) )
75, 6pm3.2i 443 . 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 )
158, 9, 10, 11, 12, 13, 14ishlat2 30213 . 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.  ) ) ) ) )
164, 7, 15mpbir2an 888 1  |-  K  e.  HL
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   ltcplt 14400   joincjn 14403   0.cp0 14468   1.cp1 14469   CLatccla 14538   OMLcoml 30035   Atomscatm 30123   AtLatcal 30124   HLchlt 30210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-cvlat 30182  df-hlat 30211
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