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Theorem hlatmstcOLDN 30208
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 22958 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 30176 . 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 30131 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B )  ->  ( U `  { y  e.  A  |  y  .<_  X } )  =  X )
71, 6sylan 457 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   lubclub 14092   CLatccla 14229   OMLcoml 29987   Atomscatm 30075   AtLatcal 30076   HLchlt 30162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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