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Theorem atlrelat1 32881
Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 28009, 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 1039 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  AtLat )
2 atlpos 32861 . . . 4  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
31, 2syl 17 . . 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 16205 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y  .<_  X )
87ex 436 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<_  X ) )
93, 8syld3an1 1313 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<_  X ) )
10 iman 426 . . . . . . 7  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  (
p  .<_  Y  /\  -.  p  .<_  X ) )
11 ancom 452 . . . . . . 7  |-  ( ( p  .<_  Y  /\  -.  p  .<_  X )  <-> 
( -.  p  .<_  X  /\  p  .<_  Y ) )
1210, 11xchbinx 312 . . . . . 6  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
1312ralbii 2818 . . . . 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 32880 . . . . . . 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 1213 . . . . . 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 227 . . . . 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 222 . . . 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 139 . . 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 2837 . . 3  |-  ( E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y )  <->  -.  A. p  e.  A  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
2119, 20syl6ibr 231 . 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 45 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736   E.wrex 2737   class class class wbr 4401   ` cfv 5581   Basecbs 15114   lecple 15190   Posetcpo 16178   ltcplt 16179   CLatccla 16346   OMLcoml 32735   Atomscatm 32823   AtLatcal 32824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-lat 16285  df-clat 16347  df-oposet 32736  df-ol 32738  df-oml 32739  df-covers 32826  df-ats 32827  df-atl 32858
This theorem is referenced by:  cvlcvr1  32899  hlrelat1  32959
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