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Theorem ishlat3N 32932
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 32930 . 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 1050 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  K  e.  CvLat )
10 simplrl 771 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  x  e.  A )
11 simplrr 772 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  y  e.  A )
12 simpr 463 . . . . . . . 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 32922 . . . . . . . 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 1271 . . . . . . 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 2900 . . . . . 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 3739 . . . . . . . 8  |-  ( x  e.  A  ->  A  =/=  (/) )
1716ad2antrl 735 . . . . . . 7  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  (
x  e.  A  /\  y  e.  A )
)  ->  A  =/=  (/) )
18 r19.37zv 3867 . . . . . . 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 17 . . . . . 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 258 . . . . 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 2832 . . . 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 712 . . 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 643 . 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 253 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 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   A.wral 2739   E.wrex 2740   (/)c0 3733   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   Basecbs 15133   lecple 15209   ltcplt 16198   joincjn 16201   0.cp0 16295   1.cp1 16296   CLatccla 16365   OMLcoml 32753   Atomscatm 32841   CvLatclc 32843   HLchlt 32928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-preset 16185  df-poset 16203  df-plt 16216  df-lub 16232  df-glb 16233  df-join 16234  df-meet 16235  df-p0 16297  df-lat 16304  df-covers 32844  df-ats 32845  df-atl 32876  df-cvlat 32900  df-hlat 32929
This theorem is referenced by: (None)
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