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Theorem atlen0 29425
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 29418 . . . . 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 29404 . . . . . 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 2387 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
133, 12, 9atcvr0 29403 . . . . . 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 2387 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
162, 15, 12cvrlt 29385 . . . . 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 29416 . . . . . 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 14355 . . . . 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 14346 . . 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 2633 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 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394   Basecbs 13396   lecple 13463   Posetcpo 14324   ltcplt 14325   0.cp0 14393    <o ccvr 29377   Atomscatm 29378   AtLatcal 29379
This theorem is referenced by:  ps-2b  29596  2atm  29641  2llnm4  29684  dalem21  29808  dalem54  29840  trlval3  30301  cdlemc5  30309
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-poset 14330  df-plt 14342  df-lat 14402  df-covers 29381  df-ats 29382  df-atl 29413
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