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Theorem ishlat1 29542
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 5525 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
2 ishlat.a . . . . . 6  |-  A  =  ( Atoms `  K )
31, 2syl6eqr 2333 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
4 fveq2 5525 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
5 ishlat.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
64, 5syl6eqr 2333 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
76breqd 4034 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( le `  k ) ( x ( join `  k
) y )  <->  z  .<_  ( x ( join `  k
) y ) ) )
8 fveq2 5525 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
9 ishlat.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
108, 9syl6eqr 2333 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1110oveqd 5875 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
x ( join `  k
) y )  =  ( x  .\/  y
) )
1211breq2d 4035 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z  .<_  ( x (
join `  k )
y )  <->  z  .<_  ( x  .\/  y ) ) )
137, 12bitrd 244 . . . . . . . . 9  |-  ( k  =  K  ->  (
z ( le `  k ) ( x ( join `  k
) y )  <->  z  .<_  ( x  .\/  y ) ) )
14133anbi3d 1258 . . . . . . . 8  |-  ( k  =  K  ->  (
( z  =/=  x  /\  z  =/=  y  /\  z ( le `  k ) ( x ( join `  k
) y ) )  <-> 
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) ) )
153, 14rexeqbidv 2749 . . . . . . 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 307 . . . . . 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 2748 . . . . 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 2748 . . . 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 5525 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
20 ishlat.b . . . . . 6  |-  B  =  ( Base `  K
)
2119, 20syl6eqr 2333 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
22 fveq2 5525 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( lt `  k )  =  ( lt `  K
) )
23 ishlat.s . . . . . . . . . . . 12  |-  .<  =  ( lt `  K )
2422, 23syl6eqr 2333 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( lt `  k )  = 
.<  )
2524breqd 4034 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  ( 0. `  k )  .<  x
) )
26 fveq2 5525 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 0. `  k )  =  ( 0. `  K
) )
27 ishlat.z . . . . . . . . . . . 12  |-  .0.  =  ( 0. `  K )
2826, 27syl6eqr 2333 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 0. `  k )  =  .0.  )
2928breq1d 4033 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
)  .<  x  <->  .0.  .<  x
) )
3025, 29bitrd 244 . . . . . . . . 9  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  .0.  .<  x
) )
3124breqd 4034 . . . . . . . . 9  |-  ( k  =  K  ->  (
x ( lt `  k ) y  <->  x  .<  y ) )
3230, 31anbi12d 691 . . . . . . . 8  |-  ( k  =  K  ->  (
( ( 0. `  k ) ( lt
`  k ) x  /\  x ( lt
`  k ) y )  <->  (  .0.  .<  x  /\  x  .<  y
) ) )
3324breqd 4034 . . . . . . . . 9  |-  ( k  =  K  ->  (
y ( lt `  k ) z  <->  y  .<  z ) )
3424breqd 4034 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( lt `  k ) ( 1.
`  k )  <->  z  .<  ( 1. `  k ) ) )
35 fveq2 5525 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 1. `  k )  =  ( 1. `  K
) )
36 ishlat.u . . . . . . . . . . . 12  |-  .1.  =  ( 1. `  K )
3735, 36syl6eqr 2333 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 1. `  k )  =  .1.  )
3837breq2d 4035 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z  .<  ( 1. `  k )  <->  z  .<  .1.  ) )
3934, 38bitrd 244 . . . . . . . . 9  |-  ( k  =  K  ->  (
z ( lt `  k ) ( 1.
`  k )  <->  z  .<  .1.  ) )
4033, 39anbi12d 691 . . . . . . . 8  |-  ( k  =  K  ->  (
( y ( lt
`  k ) z  /\  z ( lt
`  k ) ( 1. `  k ) )  <->  ( y  .< 
z  /\  z  .<  .1.  ) ) )
4132, 40anbi12d 691 . . . . . . 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 2749 . . . . . 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 2749 . . . . 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 2749 . . . 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 691 . . 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 29541 . . 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 2925 . 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 3358 . . . . 5  |-  ( K  e.  ( OML  i^i  CLat )  <->  ( K  e. 
OML  /\  K  e.  CLat ) )
4948anbi1i 676 . . . 4  |-  ( ( K  e.  ( OML 
i^i  CLat )  /\  K  e.  CvLat )  <->  ( ( K  e.  OML  /\  K  e.  CLat )  /\  K  e.  CvLat ) )
50 elin 3358 . . . 4  |-  ( K  e.  ( ( OML 
i^i  CLat )  i^i  CvLat )  <-> 
( K  e.  ( OML  i^i  CLat )  /\  K  e.  CvLat ) )
51 df-3an 936 . . . 4  |-  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  <->  ( ( K  e.  OML  /\  K  e.  CLat )  /\  K  e.  CvLat ) )
5249, 50, 513bitr4ri 269 . . 3  |-  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  <->  K  e.  ( ( OML  i^i  CLat )  i^i  CvLat ) )
5352anbi1i 676 . 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 243 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   ltcplt 14075   joincjn 14078   0.cp0 14143   1.cp1 14144   CLatccla 14213   OMLcoml 29365   Atomscatm 29453   CvLatclc 29455   HLchlt 29540
This theorem is referenced by:  ishlat2  29543  ishlat3N  29544  hlomcmcv  29546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-hlat 29541
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