Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atlen0 Unicode version

Theorem atlen0 29793
Description: A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
atlen0.b  |-  B  =  ( Base `  K
)
atlen0.l  |-  .<_  =  ( le `  K )
atlen0.z  |-  .0.  =  ( 0. `  K )
atlen0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atlen0  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  X  =/=  .0.  )

Proof of Theorem atlen0
StepHypRef Expression
1 simpl1 960 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  K  e.  AtLat )
2 atlen0.b . . . . . 6  |-  B  =  ( Base `  K
)
3 atlen0.z . . . . . 6  |-  .0.  =  ( 0. `  K )
42, 3atl0cl 29786 . . . . 5  |-  ( K  e.  AtLat  ->  .0.  e.  B )
51, 4syl 16 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  e.  B
)
6 simpl2 961 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  X  e.  B
)
71, 5, 63jca 1134 . . 3  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  ( K  e. 
AtLat  /\  .0.  e.  B  /\  X  e.  B
) )
8 simpl3 962 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  P  e.  A
)
9 atlen0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
102, 9atbase 29772 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
118, 10syl 16 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  P  e.  B
)
12 eqid 2404 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
133, 12, 9atcvr0 29771 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  (  <o  `  K ) P )
141, 8, 13syl2anc 643 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  (  <o  `  K ) P )
15 eqid 2404 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
162, 15, 12cvrlt 29753 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  .0.  e.  B  /\  P  e.  B )  /\  .0.  (  <o  `  K
) P )  ->  .0.  ( lt `  K
) P )
171, 5, 11, 14, 16syl31anc 1187 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  ( lt `  K ) P )
18 simpr 448 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  P  .<_  X )
19 atlpos 29784 . . . . . 6  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
201, 19syl 16 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  K  e.  Poset )
21 atlen0.l . . . . . 6  |-  .<_  =  ( le `  K )
222, 21, 15pltletr 14383 . . . . 5  |-  ( ( K  e.  Poset  /\  (  .0.  e.  B  /\  P  e.  B  /\  X  e.  B ) )  -> 
( (  .0.  ( lt `  K ) P  /\  P  .<_  X )  ->  .0.  ( lt `  K ) X ) )
2320, 5, 11, 6, 22syl13anc 1186 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  ( (  .0.  ( lt `  K
) P  /\  P  .<_  X )  ->  .0.  ( lt `  K ) X ) )
2417, 18, 23mp2and 661 . . 3  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  ( lt `  K ) X )
2515pltne 14374 . . 3  |-  ( ( K  e.  AtLat  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  ( lt `  K
) X  ->  .0.  =/=  X ) )
267, 24, 25sylc 58 . 2  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  =/=  X
)
2726necomd 2650 1  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  X  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Posetcpo 14352   ltcplt 14353   0.cp0 14421    <o ccvr 29745   Atomscatm 29746   AtLatcal 29747
This theorem is referenced by:  ps-2b  29964  2atm  30009  2llnm4  30052  dalem21  30176  dalem54  30208  trlval3  30669  cdlemc5  30677
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-poset 14358  df-plt 14370  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781
  Copyright terms: Public domain W3C validator