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Theorem ishlat3N 29596
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 29594 . 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 996 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  K  e.  CvLat )
10 simplrl 736 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  x  e.  A )
11 simplrr 737 . . . . . . . 8  |-  ( ( ( ( K  e. 
OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  y  e.  A )
12 simpr 447 . . . . . . . 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 29586 . . . . . . . 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 1184 . . . . . . 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 2636 . . . . . 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 3537 . . . . . . . 8  |-  ( x  e.  A  ->  A  =/=  (/) )
1716ad2antrl 708 . . . . . . 7  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  (
x  e.  A  /\  y  e.  A )
)  ->  A  =/=  (/) )
18 r19.37zv 3626 . . . . . . 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 15 . . . . . 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 245 . . . . 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 2659 . . . 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 685 . . 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 618 . 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 240 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   (/)c0 3531   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   Basecbs 13239   lecple 13306   ltcplt 14168   joincjn 14171   0.cp0 14236   1.cp1 14237   CLatccla 14306   OMLcoml 29417   Atomscatm 29505   CvLatclc 29507   HLchlt 29592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-poset 14173  df-plt 14185  df-lub 14201  df-join 14203  df-lat 14245  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593
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