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Theorem hlatmstcOLDN 29586
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 22942 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 29554 . 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 29509 . 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 1623    e. wcel 1684   {crab 2547   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   lubclub 14076   CLatccla 14213   OMLcoml 29365   Atomscatm 29453   AtLatcal 29454   HLchlt 29540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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