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Theorem atlrelat1 32800
Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 27992, 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 1037 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  AtLat )
2 atlpos 32780 . . . 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 16190 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y  .<_  X )
87ex 435 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<_  X ) )
93, 8syld3an1 1310 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  -.  Y  .<_  X ) )
10 iman 425 . . . . . . 7  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  (
p  .<_  Y  /\  -.  p  .<_  X ) )
11 ancom 451 . . . . . . 7  |-  ( ( p  .<_  Y  /\  -.  p  .<_  X )  <-> 
( -.  p  .<_  X  /\  p  .<_  Y ) )
1210, 11xchbinx 311 . . . . . 6  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
1312ralbii 2854 . . . . 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 32799 . . . . . . 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 1211 . . . . . 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 226 . . . . 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 221 . . . 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 138 . . 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 2874 . . 3  |-  ( E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y )  <->  -.  A. p  e.  A  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
2119, 20syl6ibr 230 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774   class class class wbr 4417   ` cfv 5593   Basecbs 15099   lecple 15175   Posetcpo 16163   ltcplt 16164   CLatccla 16331   OMLcoml 32654   Atomscatm 32742   AtLatcal 32743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6259  df-ov 6300  df-oprab 6301  df-preset 16151  df-poset 16169  df-plt 16182  df-lub 16198  df-glb 16199  df-join 16200  df-meet 16201  df-p0 16263  df-lat 16270  df-clat 16332  df-oposet 32655  df-ol 32657  df-oml 32658  df-covers 32745  df-ats 32746  df-atl 32777
This theorem is referenced by:  cvlcvr1  32818  hlrelat1  32878
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