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Theorem ishlat1 33002
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 5696 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
2 ishlat.a . . . . . 6  |-  A  =  ( Atoms `  K )
31, 2syl6eqr 2493 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
4 fveq2 5696 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
5 ishlat.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
64, 5syl6eqr 2493 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
76breqd 4308 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( le `  k ) ( x ( join `  k
) y )  <->  z  .<_  ( x ( join `  k
) y ) ) )
8 fveq2 5696 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
9 ishlat.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
108, 9syl6eqr 2493 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1110oveqd 6113 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
x ( join `  k
) y )  =  ( x  .\/  y
) )
1211breq2d 4309 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z  .<_  ( x (
join `  k )
y )  <->  z  .<_  ( x  .\/  y ) ) )
137, 12bitrd 253 . . . . . . . . 9  |-  ( k  =  K  ->  (
z ( le `  k ) ( x ( join `  k
) y )  <->  z  .<_  ( x  .\/  y ) ) )
14133anbi3d 1295 . . . . . . . 8  |-  ( k  =  K  ->  (
( z  =/=  x  /\  z  =/=  y  /\  z ( le `  k ) ( x ( join `  k
) y ) )  <-> 
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) ) )
153, 14rexeqbidv 2937 . . . . . . 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 316 . . . . . 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 2936 . . . . 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 2936 . . . 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 5696 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
20 ishlat.b . . . . . 6  |-  B  =  ( Base `  K
)
2119, 20syl6eqr 2493 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
22 fveq2 5696 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( lt `  k )  =  ( lt `  K
) )
23 ishlat.s . . . . . . . . . . . 12  |-  .<  =  ( lt `  K )
2422, 23syl6eqr 2493 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( lt `  k )  = 
.<  )
2524breqd 4308 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  ( 0. `  k )  .<  x
) )
26 fveq2 5696 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 0. `  k )  =  ( 0. `  K
) )
27 ishlat.z . . . . . . . . . . . 12  |-  .0.  =  ( 0. `  K )
2826, 27syl6eqr 2493 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 0. `  k )  =  .0.  )
2928breq1d 4307 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
)  .<  x  <->  .0.  .<  x
) )
3025, 29bitrd 253 . . . . . . . . 9  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  .0.  .<  x
) )
3124breqd 4308 . . . . . . . . 9  |-  ( k  =  K  ->  (
x ( lt `  k ) y  <->  x  .<  y ) )
3230, 31anbi12d 710 . . . . . . . 8  |-  ( k  =  K  ->  (
( ( 0. `  k ) ( lt
`  k ) x  /\  x ( lt
`  k ) y )  <->  (  .0.  .<  x  /\  x  .<  y
) ) )
3324breqd 4308 . . . . . . . . 9  |-  ( k  =  K  ->  (
y ( lt `  k ) z  <->  y  .<  z ) )
3424breqd 4308 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( lt `  k ) ( 1.
`  k )  <->  z  .<  ( 1. `  k ) ) )
35 fveq2 5696 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 1. `  k )  =  ( 1. `  K
) )
36 ishlat.u . . . . . . . . . . . 12  |-  .1.  =  ( 1. `  K )
3735, 36syl6eqr 2493 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 1. `  k )  =  .1.  )
3837breq2d 4309 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z  .<  ( 1. `  k )  <->  z  .<  .1.  ) )
3934, 38bitrd 253 . . . . . . . . 9  |-  ( k  =  K  ->  (
z ( lt `  k ) ( 1.
`  k )  <->  z  .<  .1.  ) )
4033, 39anbi12d 710 . . . . . . . 8  |-  ( k  =  K  ->  (
( y ( lt
`  k ) z  /\  z ( lt
`  k ) ( 1. `  k ) )  <->  ( y  .< 
z  /\  z  .<  .1.  ) ) )
4132, 40anbi12d 710 . . . . . . 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 2937 . . . . . 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 2937 . . . . 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 2937 . . . 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 710 . . 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 33001 . . 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 3124 . 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 3544 . . . . 5  |-  ( K  e.  ( OML  i^i  CLat )  <->  ( K  e. 
OML  /\  K  e.  CLat ) )
4948anbi1i 695 . . . 4  |-  ( ( K  e.  ( OML 
i^i  CLat )  /\  K  e.  CvLat )  <->  ( ( K  e.  OML  /\  K  e.  CLat )  /\  K  e.  CvLat ) )
50 elin 3544 . . . 4  |-  ( K  e.  ( ( OML 
i^i  CLat )  i^i  CvLat )  <-> 
( K  e.  ( OML  i^i  CLat )  /\  K  e.  CvLat ) )
51 df-3an 967 . . . 4  |-  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  <->  ( ( K  e.  OML  /\  K  e.  CLat )  /\  K  e.  CvLat ) )
5249, 50, 513bitr4ri 278 . . 3  |-  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  <->  K  e.  ( ( OML  i^i  CLat )  i^i  CvLat ) )
5352anbi1i 695 . 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 252 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721    i^i cin 3332   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   ltcplt 15116   joincjn 15119   0.cp0 15212   1.cp1 15213   CLatccla 15282   OMLcoml 32825   Atomscatm 32913   CvLatclc 32915   HLchlt 33000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431  df-ov 6099  df-hlat 33001
This theorem is referenced by:  ishlat2  33003  ishlat3N  33004  hlomcmcv  33006
  Copyright terms: Public domain W3C validator