Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atlrelat1 Unicode version

Theorem atlrelat1 28778
Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 22935, 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 28758 . . . 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 14094 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y  .<_  X )
87ex 425 . . 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 415 . . . . . . 7  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  (
p  .<_  Y  /\  -.  p  .<_  X ) )
11 ancom 439 . . . . . . 7  |-  ( ( p  .<_  Y  /\  -.  p  .<_  X )  <-> 
( -.  p  .<_  X  /\  p  .<_  Y ) )
1210, 11xchbinx 303 . . . . . 6  |-  ( ( p  .<_  Y  ->  p 
.<_  X )  <->  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
1312ralbii 2568 . . . . 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 28777 . . . . . . 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 216 . . . . 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 211 . . . 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 2557 . . 3  |-  ( E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y )  <->  -.  A. p  e.  A  -.  ( -.  p  .<_  X  /\  p  .<_  Y ) )
2119, 20syl6ibr 220 . 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   A.wral 2544   E.wrex 2545   class class class wbr 4024   ` cfv 5221   Basecbs 13142   lecple 13209   Posetcpo 14068   ltcplt 14069   CLatccla 14207   OMLcoml 28632   Atomscatm 28720   AtLatcal 28721
This theorem is referenced by:  cvlcvr1  28796  hlrelat1  28856
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755
  Copyright terms: Public domain W3C validator