Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hlexch4N Unicode version

Theorem hlexch4N 28831
Description: A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlexch3.b  |-  B  =  ( Base `  K
)
hlexch3.l  |-  .<_  =  ( le `  K )
hlexch3.j  |-  .\/  =  ( join `  K )
hlexch3.m  |-  ./\  =  ( meet `  K )
hlexch3.z  |-  .0.  =  ( 0. `  K )
hlexch3.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlexch4N  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  ./\ 
X )  =  .0.  )  ->  ( P  .<_  ( X  .\/  Q
)  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )

Proof of Theorem hlexch4N
StepHypRef Expression
1 hlcvl 28799 . 2  |-  ( K  e.  HL  ->  K  e.  CvLat )
2 hlexch3.b . . 3  |-  B  =  ( Base `  K
)
3 hlexch3.l . . 3  |-  .<_  =  ( le `  K )
4 hlexch3.j . . 3  |-  .\/  =  ( join `  K )
5 hlexch3.m . . 3  |-  ./\  =  ( meet `  K )
6 hlexch3.z . . 3  |-  .0.  =  ( 0. `  K )
7 hlexch3.a . . 3  |-  A  =  ( Atoms `  K )
82, 3, 4, 5, 6, 7cvlexch4N 28773 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  .<_  ( X 
.\/  Q )  <->  ( X  .\/  P )  =  ( X  .\/  Q ) ) )
91, 8syl3an1 1220 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  ./\ 
X )  =  .0.  )  ->  ( P  .<_  ( X  .\/  Q
)  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   0.cp0 14106   Atomscatm 28703   CvLatclc 28705   HLchlt 28790
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-lat 14115  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791
  Copyright terms: Public domain W3C validator