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Theorem ishlat3N 30250
Description: The predicate "is a Hilbert lattice". Note that the superposition principle is expressed in the compact form  E. z  e.  A ( x  .\/  z )  =  ( y  .\/  z ). The exchange property and atomicity are provided by  K  e.  CvLat, and "minimum height 4" is shown explicitly. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.)
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
ishlat3N  |-  ( K  e.  HL  <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
)  /\  ( A. x  e.  A  A. y  e.  A  E. z  e.  A  (
x  .\/  z )  =  ( y  .\/  z )  /\  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 ishlat3N
StepHypRef Expression
1 ishlat.b . . 3  |-  B  =  ( Base `  K
)
2 ishlat.l . . 3  |-  .<_  =  ( le `  K )
3 ishlat.s . . 3  |-  .<  =  ( lt `  K )
4 ishlat.j . . 3  |-  .\/  =  ( join `  K )
5 ishlat.z . . 3  |-  .0.  =  ( 0. `  K )
6 ishlat.u . . 3  |-  .1.  =  ( 1. `  K )
7 ishlat.a . . 3  |-  A  =  ( Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat1 30248 . 2  |-  ( 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.  ) ) ) ) )
9 simpll3 999 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  K  e.  CvLat )
10 simplrl 738 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  x  e.  A )
11 simplrr 739 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  y  e.  A )
12 simpr 449 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  z  e.  A )
137, 2, 4cvlsupr3 30240 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  ->  ( (
x  .\/  z )  =  ( y  .\/  z )  <->  ( x  =/=  y  ->  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y
) ) ) ) )
149, 10, 11, 12, 13syl13anc 1187 . . . . . . 7  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
( x  .\/  z
)  =  ( y 
.\/  z )  <->  ( x  =/=  y  ->  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y
) ) ) ) )
1514rexbidva 2728 . . . . . 6  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  (
x  e.  A  /\  y  e.  A )
)  ->  ( E. z  e.  A  (
x  .\/  z )  =  ( y  .\/  z )  <->  E. z  e.  A  ( x  =/=  y  ->  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y
) ) ) ) )
16 ne0i 3619 . . . . . . . 8  |-  ( x  e.  A  ->  A  =/=  (/) )
1716ad2antrl 710 . . . . . . 7  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  (
x  e.  A  /\  y  e.  A )
)  ->  A  =/=  (/) )
18 r19.37zv 3748 . . . . . . 7  |-  ( A  =/=  (/)  ->  ( E. z  e.  A  (
x  =/=  y  -> 
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  <->  ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y ) ) ) ) )
1917, 18syl 16 . . . . . 6  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  (
x  e.  A  /\  y  e.  A )
)  ->  ( E. z  e.  A  (
x  =/=  y  -> 
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  <->  ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y ) ) ) ) )
2015, 19bitr2d 247 . . . . 5  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  (
x  e.  A  /\  y  e.  A )
)  ->  ( (
x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  <->  E. z  e.  A  ( x  .\/  z )  =  ( y  .\/  z ) ) )
21202ralbidva 2751 . . . 4  |-  ( ( 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 ) ) )  <->  A. x  e.  A  A. y  e.  A  E. z  e.  A  ( x  .\/  z )  =  ( y  .\/  z ) ) )
2221anbi1d 687 . . 3  |-  ( ( 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.  ) ) )  <->  ( A. x  e.  A  A. y  e.  A  E. z  e.  A  (
x  .\/  z )  =  ( y  .\/  z )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  (
(  .0.  .<  x  /\  x  .<  y )  /\  ( y  .< 
z  /\  z  .<  .1.  ) ) ) ) )
2322pm5.32i 620 . 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  /\  K  e.  CLat  /\  K  e.  CvLat
)  /\  ( A. x  e.  A  A. y  e.  A  E. z  e.  A  (
x  .\/  z )  =  ( y  .\/  z )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  (
(  .0.  .<  x  /\  x  .<  y )  /\  ( y  .< 
z  /\  z  .<  .1.  ) ) ) ) )
248, 23bitri 242 1  |-  ( K  e.  HL  <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
)  /\  ( A. x  e.  A  A. y  e.  A  E. z  e.  A  (
x  .\/  z )  =  ( y  .\/  z )  /\  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 1727    =/= wne 2605   A.wral 2711   E.wrex 2712   (/)c0 3613   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lecple 13567   ltcplt 14429   joincjn 14432   0.cp0 14497   1.cp1 14498   CLatccla 14567   OMLcoml 30071   Atomscatm 30159   CvLatclc 30161   HLchlt 30246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
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-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-plt 14446  df-lub 14462  df-join 14464  df-lat 14506  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247
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