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Theorem atnem0 29813
Description: The meet of distinct atoms is zero. (atnemeq0 23841 analog.) (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
atnem0.m  |-  ./\  =  ( meet `  K )
atnem0.z  |-  .0.  =  ( 0. `  K )
atnem0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atnem0  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  ( P  ./\ 
Q )  =  .0.  ) )

Proof of Theorem atnem0
StepHypRef Expression
1 eqid 2412 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 atnem0.a . . 3  |-  A  =  ( Atoms `  K )
31, 2atncmp 29807 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( -.  P ( le `  K ) Q  <->  P  =/=  Q ) )
4 eqid 2412 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
54, 2atbase 29784 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
6 atnem0.m . . . 4  |-  ./\  =  ( meet `  K )
7 atnem0.z . . . 4  |-  .0.  =  ( 0. `  K )
84, 1, 6, 7, 2atnle 29812 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  ( Base `  K
) )  ->  ( -.  P ( le `  K ) Q  <->  ( P  ./\ 
Q )  =  .0.  ) )
95, 8syl3an3 1219 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( -.  P ( le `  K ) Q  <->  ( P  ./\ 
Q )  =  .0.  ) )
103, 9bitr3d 247 1  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  ( P  ./\ 
Q )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   meetcmee 14365   0.cp0 14429   Atomscatm 29758   AtLatcal 29759
This theorem is referenced by:  cvlatcvr1  29836  atcvrj1  29925  dalem24  30191  lhp2at0  30526  trlval3  30681  cdleme0e  30711  cdleme7c  30739
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-glb 14395  df-meet 14397  df-p0 14431  df-lat 14438  df-covers 29761  df-ats 29762  df-atl 29793
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