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Theorem atlrelat1 29511
Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 22943, with  /\ swapped, analog.) (Contributed by NM, 4-Dec-2011.)
Hypotheses
Ref Expression
atlrelat1.b  |-  B  =  ( Base `  K
)
atlrelat1.l  |-  .<_  =  ( le `  K )
atlrelat1.s  |-  .<  =  ( lt `  K )
atlrelat1.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atlrelat1  |-  ( ( ( 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 ) ) )
Distinct variable groups:    A, p    B, p    K, p    .<_ , p    X, p    Y, p
Allowed substitution hint:    .< ( p)

Proof of Theorem atlrelat1
StepHypRef Expression
1 simp13 987 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  AtLat )
2 atlpos 29491 . . . 4  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
31, 2syl 15 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
4 atlrelat1.b . . . . 5  |-  B  =  ( Base `  K
)
5 atlrelat1.l . . . . 5  |-  .<_  =  ( le `  K )
6 atlrelat1.s . . . . 5  |-  .<  =  ( lt `  K )
74, 5, 6pltnle 14100 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y  .<_  X )
87ex 423 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<_  X ) )
93, 8syld3an1 1228 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<_  X ) )
10 iman 413 . . . . . . 7  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  (
p  .<_  Y  /\  -.  p  .<_  X ) )
11 ancom 437 . . . . . . 7  |-  ( ( p  .<_  Y  /\  -.  p  .<_  X )  <-> 
( -.  p  .<_  X  /\  p  .<_  Y ) )
1210, 11xchbinx 301 . . . . . 6  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
1312ralbii 2567 . . . . 5  |-  ( A. p  e.  A  (
p  .<_  Y  ->  p  .<_  X )  <->  A. p  e.  A  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
14 atlrelat1.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
154, 5, 14atlatle 29510 . . . . . . 7  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<_  X  <->  A. p  e.  A  ( p  .<_  Y  ->  p  .<_  X ) ) )
16153com23 1157 . . . . . 6  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  A. p  e.  A  ( p  .<_  Y  ->  p  .<_  X ) ) )
1716biimprd 214 . . . . 5  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( A. p  e.  A  ( p  .<_  Y  ->  p  .<_  X )  ->  Y  .<_  X ) )
1813, 17syl5bir 209 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( A. p  e.  A  -.  ( -.  p  .<_  X  /\  p  .<_  Y )  ->  Y  .<_  X ) )
1918con3d 125 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  ->  -.  A. p  e.  A  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
20 dfrex2 2556 . . 3  |-  ( E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y )  <->  -.  A. p  e.  A  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
2119, 20syl6ibr 218 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
229, 21syld 40 1  |-  ( ( ( 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 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075   CLatccla 14213   OMLcoml 29365   Atomscatm 29453   AtLatcal 29454
This theorem is referenced by:  cvlcvr1  29529  hlrelat1  29589
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
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