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Theorem ishlat3N 35222
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 35220 . 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 1037 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  K  e.  CvLat )
10 simplrl 761 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  x  e.  A )
11 simplrr 762 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  y  e.  A )
12 simpr 461 . . . . . . . 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 35212 . . . . . . . 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 1230 . . . . . . 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 2965 . . . . . 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 3799 . . . . . . . 8  |-  ( x  e.  A  ->  A  =/=  (/) )
1716ad2antrl 727 . . . . . . 7  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  (
x  e.  A  /\  y  e.  A )
)  ->  A  =/=  (/) )
18 r19.37zv 3928 . . . . . . 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 254 . . . . 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 2899 . . . 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 704 . . 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 637 . 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 249 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   (/)c0 3793   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   lecple 14719   ltcplt 15697   joincjn 15700   0.cp0 15794   1.cp1 15795   CLatccla 15864   OMLcoml 35043   Atomscatm 35131   CvLatclc 35133   HLchlt 35218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-lat 15803  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219
This theorem is referenced by: (None)
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