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Theorem atlen0 30122
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 30115 . . . . 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 30101 . . . . . 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 2296 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
133, 12, 9atcvr0 30100 . . . . . 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 2296 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
162, 15, 12cvrlt 30082 . . . . 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 30113 . . . . . 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 14121 . . . . 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 14112 . . 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 2542 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   ltcplt 14091   0.cp0 14159    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076
This theorem is referenced by:  ps-2b  30293  2atm  30338  2llnm4  30381  dalem21  30505  dalem54  30537  trlval3  30998  cdlemc5  31006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-poset 14096  df-plt 14108  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110
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