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Theorem hlatmstcOLDN 29511
Description: An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 23713 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlatmstc.b  |-  B  =  ( Base `  K
)
hlatmstc.l  |-  .<_  =  ( le `  K )
hlatmstc.u  |-  U  =  ( lub `  K
)
hlatmstc.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlatmstcOLDN  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( U `  {
y  e.  A  | 
y  .<_  X } )  =  X )
Distinct variable groups:    y, A    y, B    y,  .<_    y, X
Allowed substitution hints:    U( y)    K( y)

Proof of Theorem hlatmstcOLDN
StepHypRef Expression
1 hlomcmat 29479 . 2  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
2 hlatmstc.b . . 3  |-  B  =  ( Base `  K
)
3 hlatmstc.l . . 3  |-  .<_  =  ( le `  K )
4 hlatmstc.u . . 3  |-  U  =  ( lub `  K
)
5 hlatmstc.a . . 3  |-  A  =  ( Atoms `  K )
62, 3, 4, 5atlatmstc 29434 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B )  ->  ( U `  { y  e.  A  |  y  .<_  X } )  =  X )
71, 6sylan 458 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( U `  {
y  e.  A  | 
y  .<_  X } )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2653   class class class wbr 4153   ` cfv 5394   Basecbs 13396   lecple 13463   lubclub 14326   CLatccla 14463   OMLcoml 29290   Atomscatm 29378   AtLatcal 29379   HLchlt 29465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466
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