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Theorem atlen0 29500
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 958 . . . 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 29493 . . . . 5  |-  ( K  e.  AtLat  ->  .0.  e.  B )
51, 4syl 15 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  e.  B
)
6 simpl2 959 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  X  e.  B
)
71, 5, 63jca 1132 . . 3  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  ( K  e. 
AtLat  /\  .0.  e.  B  /\  X  e.  B
) )
8 simpl3 960 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  P  e.  A
)
9 atlen0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
102, 9atbase 29479 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
118, 10syl 15 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  P  e.  B
)
12 eqid 2283 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
133, 12, 9atcvr0 29478 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  (  <o  `  K ) P )
141, 8, 13syl2anc 642 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  (  <o  `  K ) P )
15 eqid 2283 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
162, 15, 12cvrlt 29460 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  .0.  e.  B  /\  P  e.  B )  /\  .0.  (  <o  `  K
) P )  ->  .0.  ( lt `  K
) P )
171, 5, 11, 14, 16syl31anc 1185 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  ( lt `  K ) P )
18 simpr 447 . . . 4  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  P  .<_  X )
19 atlpos 29491 . . . . . 6  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
201, 19syl 15 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  K  e.  Poset )
21 atlen0.l . . . . . 6  |-  .<_  =  ( le `  K )
222, 21, 15pltletr 14105 . . . . 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 1184 . . . 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 660 . . 3  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  ( lt `  K ) X )
2515pltne 14096 . . 3  |-  ( ( K  e.  AtLat  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  ( lt `  K
) X  ->  .0.  =/=  X ) )
267, 24, 25sylc 56 . 2  |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  .0.  =/=  X
)
2726necomd 2529 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075   0.cp0 14143    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454
This theorem is referenced by:  ps-2b  29671  2atm  29716  2llnm4  29759  dalem21  29883  dalem54  29915  trlval3  30376  cdlemc5  30384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488
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