Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ishlat1 Structured version   Unicode version

Theorem ishlat1 30151
Description: The predicate "is a Hilbert lattice," which is orthomodular ( K  e.  OML), complete ( K  e.  CLat), atomic and satisfying the exchange (or covering) property ( K  e.  CvLat), satisfies the superposition principle, and has a minimum height of 4. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
ishlat.b  |-  B  =  ( Base `  K
)
ishlat.l  |-  .<_  =  ( le `  K )
ishlat.s  |-  .<  =  ( lt `  K )
ishlat.j  |-  .\/  =  ( join `  K )
ishlat.z  |-  .0.  =  ( 0. `  K )
ishlat.u  |-  .1.  =  ( 1. `  K )
ishlat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
ishlat1  |-  ( K  e.  HL  <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
)  /\  ( A. x  e.  A  A. y  e.  A  (
x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) ) )
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 ishlat1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fveq2 5729 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
2 ishlat.a . . . . . 6  |-  A  =  ( Atoms `  K )
31, 2syl6eqr 2487 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
4 fveq2 5729 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
5 ishlat.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
64, 5syl6eqr 2487 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
76breqd 4224 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( le `  k ) ( x ( join `  k
) y )  <->  z  .<_  ( x ( join `  k
) y ) ) )
8 fveq2 5729 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
9 ishlat.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
108, 9syl6eqr 2487 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1110oveqd 6099 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
x ( join `  k
) y )  =  ( x  .\/  y
) )
1211breq2d 4225 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z  .<_  ( x (
join `  k )
y )  <->  z  .<_  ( x  .\/  y ) ) )
137, 12bitrd 246 . . . . . . . . 9  |-  ( k  =  K  ->  (
z ( le `  k ) ( x ( join `  k
) y )  <->  z  .<_  ( x  .\/  y ) ) )
14133anbi3d 1261 . . . . . . . 8  |-  ( k  =  K  ->  (
( z  =/=  x  /\  z  =/=  y  /\  z ( le `  k ) ( x ( join `  k
) y ) )  <-> 
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) ) )
153, 14rexeqbidv 2918 . . . . . . 7  |-  ( k  =  K  ->  ( E. z  e.  ( Atoms `  k ) ( z  =/=  x  /\  z  =/=  y  /\  z
( le `  k
) ( x (
join `  k )
y ) )  <->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y ) ) ) )
1615imbi2d 309 . . . . . 6  |-  ( k  =  K  ->  (
( x  =/=  y  ->  E. z  e.  (
Atoms `  k ) ( z  =/=  x  /\  z  =/=  y  /\  z
( le `  k
) ( x (
join `  k )
y ) ) )  <-> 
( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) ) ) )
173, 16raleqbidv 2917 . . . . 5  |-  ( k  =  K  ->  ( A. y  e.  ( Atoms `  k ) ( x  =/=  y  ->  E. z  e.  ( Atoms `  k ) ( z  =/=  x  /\  z  =/=  y  /\  z
( le `  k
) ( x (
join `  k )
y ) ) )  <->  A. y  e.  A  ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) ) ) )
183, 17raleqbidv 2917 . . . 4  |-  ( k  =  K  ->  ( A. x  e.  ( Atoms `  k ) A. y  e.  ( Atoms `  k ) ( x  =/=  y  ->  E. z  e.  ( Atoms `  k )
( z  =/=  x  /\  z  =/=  y  /\  z ( le `  k ) ( x ( join `  k
) y ) ) )  <->  A. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) ) ) )
19 fveq2 5729 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
20 ishlat.b . . . . . 6  |-  B  =  ( Base `  K
)
2119, 20syl6eqr 2487 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
22 fveq2 5729 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( lt `  k )  =  ( lt `  K
) )
23 ishlat.s . . . . . . . . . . . 12  |-  .<  =  ( lt `  K )
2422, 23syl6eqr 2487 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( lt `  k )  = 
.<  )
2524breqd 4224 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  ( 0. `  k )  .<  x
) )
26 fveq2 5729 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 0. `  k )  =  ( 0. `  K
) )
27 ishlat.z . . . . . . . . . . . 12  |-  .0.  =  ( 0. `  K )
2826, 27syl6eqr 2487 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 0. `  k )  =  .0.  )
2928breq1d 4223 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
)  .<  x  <->  .0.  .<  x
) )
3025, 29bitrd 246 . . . . . . . . 9  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  .0.  .<  x
) )
3124breqd 4224 . . . . . . . . 9  |-  ( k  =  K  ->  (
x ( lt `  k ) y  <->  x  .<  y ) )
3230, 31anbi12d 693 . . . . . . . 8  |-  ( k  =  K  ->  (
( ( 0. `  k ) ( lt
`  k ) x  /\  x ( lt
`  k ) y )  <->  (  .0.  .<  x  /\  x  .<  y
) ) )
3324breqd 4224 . . . . . . . . 9  |-  ( k  =  K  ->  (
y ( lt `  k ) z  <->  y  .<  z ) )
3424breqd 4224 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( lt `  k ) ( 1.
`  k )  <->  z  .<  ( 1. `  k ) ) )
35 fveq2 5729 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 1. `  k )  =  ( 1. `  K
) )
36 ishlat.u . . . . . . . . . . . 12  |-  .1.  =  ( 1. `  K )
3735, 36syl6eqr 2487 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 1. `  k )  =  .1.  )
3837breq2d 4225 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z  .<  ( 1. `  k )  <->  z  .<  .1.  ) )
3934, 38bitrd 246 . . . . . . . . 9  |-  ( k  =  K  ->  (
z ( lt `  k ) ( 1.
`  k )  <->  z  .<  .1.  ) )
4033, 39anbi12d 693 . . . . . . . 8  |-  ( k  =  K  ->  (
( y ( lt
`  k ) z  /\  z ( lt
`  k ) ( 1. `  k ) )  <->  ( y  .< 
z  /\  z  .<  .1.  ) ) )
4132, 40anbi12d 693 . . . . . . 7  |-  ( k  =  K  ->  (
( ( ( 0.
`  k ) ( lt `  k ) x  /\  x ( lt `  k ) y )  /\  (
y ( lt `  k ) z  /\  z ( lt `  k ) ( 1.
`  k ) ) )  <->  ( (  .0. 
.<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) )
4221, 41rexeqbidv 2918 . . . . . 6  |-  ( k  =  K  ->  ( E. z  e.  ( Base `  k ) ( ( ( 0. `  k ) ( lt
`  k ) x  /\  x ( lt
`  k ) y )  /\  ( y ( lt `  k
) z  /\  z
( lt `  k
) ( 1. `  k ) ) )  <->  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y
)  /\  ( y  .<  z  /\  z  .<  .1.  ) ) ) )
4321, 42rexeqbidv 2918 . . . . 5  |-  ( k  =  K  ->  ( E. y  e.  ( Base `  k ) E. z  e.  ( Base `  k ) ( ( ( 0. `  k
) ( lt `  k ) x  /\  x ( lt `  k ) y )  /\  ( y ( lt `  k ) z  /\  z ( lt `  k ) ( 1. `  k
) ) )  <->  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) )
4421, 43rexeqbidv 2918 . . . 4  |-  ( k  =  K  ->  ( E. x  e.  ( Base `  k ) E. y  e.  ( Base `  k ) E. z  e.  ( Base `  k
) ( ( ( 0. `  k ) ( lt `  k
) x  /\  x
( lt `  k
) y )  /\  ( y ( lt
`  k ) z  /\  z ( lt
`  k ) ( 1. `  k ) ) )  <->  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) )
4518, 44anbi12d 693 . . 3  |-  ( k  =  K  ->  (
( A. x  e.  ( Atoms `  k ) A. y  e.  ( Atoms `  k ) ( x  =/=  y  ->  E. z  e.  ( Atoms `  k ) ( z  =/=  x  /\  z  =/=  y  /\  z
( le `  k
) ( x (
join `  k )
y ) ) )  /\  E. x  e.  ( Base `  k
) E. y  e.  ( Base `  k
) E. z  e.  ( Base `  k
) ( ( ( 0. `  k ) ( lt `  k
) x  /\  x
( lt `  k
) y )  /\  ( y ( lt
`  k ) z  /\  z ( lt
`  k ) ( 1. `  k ) ) ) )  <->  ( A. x  e.  A  A. y  e.  A  (
x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) ) )
46 df-hlat 30150 . . 3  |-  HL  =  { k  e.  ( ( OML  i^i  CLat )  i^i  CvLat )  |  ( A. x  e.  (
Atoms `  k ) A. y  e.  ( Atoms `  k ) ( x  =/=  y  ->  E. z  e.  ( Atoms `  k )
( z  =/=  x  /\  z  =/=  y  /\  z ( le `  k ) ( x ( join `  k
) y ) ) )  /\  E. x  e.  ( Base `  k
) E. y  e.  ( Base `  k
) E. z  e.  ( Base `  k
) ( ( ( 0. `  k ) ( lt `  k
) x  /\  x
( lt `  k
) y )  /\  ( y ( lt
`  k ) z  /\  z ( lt
`  k ) ( 1. `  k ) ) ) ) }
4745, 46elrab2 3095 . 2  |-  ( K  e.  HL  <->  ( K  e.  ( ( OML  i^i  CLat )  i^i  CvLat )  /\  ( A. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) ) )
48 elin 3531 . . . . 5  |-  ( K  e.  ( OML  i^i  CLat )  <->  ( K  e. 
OML  /\  K  e.  CLat ) )
4948anbi1i 678 . . . 4  |-  ( ( K  e.  ( OML 
i^i  CLat )  /\  K  e.  CvLat )  <->  ( ( K  e.  OML  /\  K  e.  CLat )  /\  K  e.  CvLat ) )
50 elin 3531 . . . 4  |-  ( K  e.  ( ( OML 
i^i  CLat )  i^i  CvLat )  <-> 
( K  e.  ( OML  i^i  CLat )  /\  K  e.  CvLat ) )
51 df-3an 939 . . . 4  |-  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  <->  ( ( K  e.  OML  /\  K  e.  CLat )  /\  K  e.  CvLat ) )
5249, 50, 513bitr4ri 271 . . 3  |-  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  <->  K  e.  ( ( OML  i^i  CLat )  i^i  CvLat ) )
5352anbi1i 678 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( A. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) )  <->  ( K  e.  ( ( OML  i^i  CLat )  i^i  CvLat )  /\  ( A. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) ) )
5447, 53bitr4i 245 1  |-  ( K  e.  HL  <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
)  /\  ( A. x  e.  A  A. y  e.  A  (
x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707    i^i cin 3320   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   ltcplt 14399   joincjn 14402   0.cp0 14467   1.cp1 14468   CLatccla 14537   OMLcoml 29974   Atomscatm 30062   CvLatclc 30064   HLchlt 30149
This theorem is referenced by:  ishlat2  30152  ishlat3N  30153  hlomcmcv  30155
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 2418
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085  df-hlat 30150
  Copyright terms: Public domain W3C validator