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Theorem ishlat1 34442
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 5871 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
2 ishlat.a . . . . . 6  |-  A  =  ( Atoms `  K )
31, 2syl6eqr 2526 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
4 fveq2 5871 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
5 ishlat.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
64, 5syl6eqr 2526 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
76breqd 4463 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( le `  k ) ( x ( join `  k
) y )  <->  z  .<_  ( x ( join `  k
) y ) ) )
8 fveq2 5871 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
9 ishlat.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
108, 9syl6eqr 2526 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1110oveqd 6311 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
x ( join `  k
) y )  =  ( x  .\/  y
) )
1211breq2d 4464 . . . . . . . . . 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 1305 . . . . . . . 8  |-  ( k  =  K  ->  (
( z  =/=  x  /\  z  =/=  y  /\  z ( le `  k ) ( x ( join `  k
) y ) )  <-> 
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) ) )
153, 14rexeqbidv 3078 . . . . . . 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 3077 . . . . 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 3077 . . . 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 5871 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
20 ishlat.b . . . . . 6  |-  B  =  ( Base `  K
)
2119, 20syl6eqr 2526 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
22 fveq2 5871 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( lt `  k )  =  ( lt `  K
) )
23 ishlat.s . . . . . . . . . . . 12  |-  .<  =  ( lt `  K )
2422, 23syl6eqr 2526 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( lt `  k )  = 
.<  )
2524breqd 4463 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  ( 0. `  k )  .<  x
) )
26 fveq2 5871 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 0. `  k )  =  ( 0. `  K
) )
27 ishlat.z . . . . . . . . . . . 12  |-  .0.  =  ( 0. `  K )
2826, 27syl6eqr 2526 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 0. `  k )  =  .0.  )
2928breq1d 4462 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
)  .<  x  <->  .0.  .<  x
) )
3025, 29bitrd 253 . . . . . . . . 9  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  .0.  .<  x
) )
3124breqd 4463 . . . . . . . . 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 4463 . . . . . . . . 9  |-  ( k  =  K  ->  (
y ( lt `  k ) z  <->  y  .<  z ) )
3424breqd 4463 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( lt `  k ) ( 1.
`  k )  <->  z  .<  ( 1. `  k ) ) )
35 fveq2 5871 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 1. `  k )  =  ( 1. `  K
) )
36 ishlat.u . . . . . . . . . . . 12  |-  .1.  =  ( 1. `  K )
3735, 36syl6eqr 2526 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 1. `  k )  =  .1.  )
3837breq2d 4464 . . . . . . . . . 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 3078 . . . . . 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 3078 . . . . 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 3078 . . . 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 34441 . . 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 3268 . 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 3692 . . . . 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 3692 . . . 4  |-  ( K  e.  ( ( OML 
i^i  CLat )  i^i  CvLat )  <-> 
( K  e.  ( OML  i^i  CLat )  /\  K  e.  CvLat ) )
51 df-3an 975 . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    i^i cin 3480   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   Basecbs 14502   lecple 14574   ltcplt 15440   joincjn 15443   0.cp0 15536   1.cp1 15537   CLatccla 15606   OMLcoml 34265   Atomscatm 34353   CvLatclc 34355   HLchlt 34440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-iota 5556  df-fv 5601  df-ov 6297  df-hlat 34441
This theorem is referenced by:  ishlat2  34443  ishlat3N  34444  hlomcmcv  34446
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