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Theorem ishlat1 30164
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 5541 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
2 ishlat.a . . . . . 6  |-  A  =  ( Atoms `  K )
31, 2syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
4 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
5 ishlat.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
64, 5syl6eqr 2346 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
76breqd 4050 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( le `  k ) ( x ( join `  k
) y )  <->  z  .<_  ( x ( join `  k
) y ) ) )
8 fveq2 5541 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
9 ishlat.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
108, 9syl6eqr 2346 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1110oveqd 5891 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
x ( join `  k
) y )  =  ( x  .\/  y
) )
1211breq2d 4051 . . . . . . . . . 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 2762 . . . . . . 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 2761 . . . . 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 2761 . . . 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 5541 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
20 ishlat.b . . . . . 6  |-  B  =  ( Base `  K
)
2119, 20syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
22 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( lt `  k )  =  ( lt `  K
) )
23 ishlat.s . . . . . . . . . . . 12  |-  .<  =  ( lt `  K )
2422, 23syl6eqr 2346 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( lt `  k )  = 
.<  )
2524breqd 4050 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  ( 0. `  k )  .<  x
) )
26 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 0. `  k )  =  ( 0. `  K
) )
27 ishlat.z . . . . . . . . . . . 12  |-  .0.  =  ( 0. `  K )
2826, 27syl6eqr 2346 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 0. `  k )  =  .0.  )
2928breq1d 4049 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( 0. `  k
)  .<  x  <->  .0.  .<  x
) )
3025, 29bitrd 244 . . . . . . . . 9  |-  ( k  =  K  ->  (
( 0. `  k
) ( lt `  k ) x  <->  .0.  .<  x
) )
3124breqd 4050 . . . . . . . . 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 4050 . . . . . . . . 9  |-  ( k  =  K  ->  (
y ( lt `  k ) z  <->  y  .<  z ) )
3424breqd 4050 . . . . . . . . . 10  |-  ( k  =  K  ->  (
z ( lt `  k ) ( 1.
`  k )  <->  z  .<  ( 1. `  k ) ) )
35 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 1. `  k )  =  ( 1. `  K
) )
36 ishlat.u . . . . . . . . . . . 12  |-  .1.  =  ( 1. `  K )
3735, 36syl6eqr 2346 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( 1. `  k )  =  .1.  )
3837breq2d 4051 . . . . . . . . . 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 2762 . . . . . 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 2762 . . . . 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 2762 . . . 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 30163 . . 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 2938 . 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 3371 . . . . 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 3371 . . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    i^i cin 3164   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   ltcplt 14091   joincjn 14094   0.cp0 14159   1.cp1 14160   CLatccla 14229   OMLcoml 29987   Atomscatm 30075   CvLatclc 30077   HLchlt 30162
This theorem is referenced by:  ishlat2  30165  ishlat3N  30166  hlomcmcv  30168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-hlat 30163
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