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Theorem atlen0 30108
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 30101 . . . . 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 30087 . . . . . 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 2436 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
133, 12, 9atcvr0 30086 . . . . . 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 2436 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
162, 15, 12cvrlt 30068 . . . . 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 30099 . . . . . 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 14428 . . . . 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 14419 . . 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 2687 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 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Posetcpo 14397   ltcplt 14398   0.cp0 14466    <o ccvr 30060   Atomscatm 30061   AtLatcal 30062
This theorem is referenced by:  ps-2b  30279  2atm  30324  2llnm4  30367  dalem21  30491  dalem54  30523  trlval3  30984  cdlemc5  30992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-poset 14403  df-plt 14415  df-lat 14475  df-covers 30064  df-ats 30065  df-atl 30096
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