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Theorem atlrelat1 29438
Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 23716, 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 989 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  AtLat )
2 atlpos 29418 . . . 4  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
31, 2syl 16 . . 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 14352 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y  .<_  X )
87ex 424 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<_  X ) )
93, 8syld3an1 1230 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<_  X ) )
10 iman 414 . . . . . . 7  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  (
p  .<_  Y  /\  -.  p  .<_  X ) )
11 ancom 438 . . . . . . 7  |-  ( ( p  .<_  Y  /\  -.  p  .<_  X )  <-> 
( -.  p  .<_  X  /\  p  .<_  Y ) )
1210, 11xchbinx 302 . . . . . 6  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
1312ralbii 2675 . . . . 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 29437 . . . . . . 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 1159 . . . . . 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 215 . . . . 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 210 . . . 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 127 . . 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 2664 . . 3  |-  ( E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y )  <->  -.  A. p  e.  A  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
2119, 20syl6ibr 219 . 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 42 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652   class class class wbr 4155   ` cfv 5396   Basecbs 13398   lecple 13465   Posetcpo 14326   ltcplt 14327   CLatccla 14465   OMLcoml 29292   Atomscatm 29380   AtLatcal 29381
This theorem is referenced by:  cvlcvr1  29456  hlrelat1  29516
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415
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