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Theorem hlrelat1 30135
Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 23859, with  /\ swapped, analog.) (Contributed by NM, 4-Dec-2011.)
Hypotheses
Ref Expression
hlrelat1.b  |-  B  =  ( Base `  K
)
hlrelat1.l  |-  .<_  =  ( le `  K )
hlrelat1.s  |-  .<  =  ( lt `  K )
hlrelat1.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlrelat1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
Distinct variable groups:    A, p    B, p    K, p    .<_ , p    X, p    Y, p
Allowed substitution hint:    .< ( p)

Proof of Theorem hlrelat1
StepHypRef Expression
1 hlomcmat 30100 . 2  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
2 hlrelat1.b . . 3  |-  B  =  ( Base `  K
)
3 hlrelat1.l . . 3  |-  .<_  =  ( le `  K )
4 hlrelat1.s . . 3  |-  .<  =  ( lt `  K )
5 hlrelat1.a . . 3  |-  A  =  ( Atoms `  K )
62, 3, 4, 5atlrelat1 30057 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
71, 6syl3an1 1217 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2699   class class class wbr 4205   ` cfv 5447   Basecbs 13462   lecple 13529   ltcplt 14391   CLatccla 14529   OMLcoml 29911   Atomscatm 29999   AtLatcal 30000   HLchlt 30086
This theorem is referenced by:  hlrelat5N  30136  hlrelat  30137  hl2at  30140  hlrelat3  30147  cvrexchlem  30154  lhpexle3lem  30746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087
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