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Theorem atle 29930
Description: Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)
Hypotheses
Ref Expression
atle.b  |-  B  =  ( Base `  K
)
atle.l  |-  .<_  =  ( le `  K )
atle.z  |-  .0.  =  ( 0. `  K )
atle.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atle  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  p  .<_  X )
Distinct variable groups:    A, p    B, p    K, p    .<_ , p    X, p    .0. , p

Proof of Theorem atle
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  K  e.  HL )
2 hlop 29857 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
323ad2ant1 978 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  K  e.  OP )
4 atle.b . . . . 5  |-  B  =  ( Base `  K
)
5 atle.z . . . . 5  |-  .0.  =  ( 0. `  K )
64, 5op0cl 29679 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
73, 6syl 16 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  .0.  e.  B )
8 simp2 958 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  e.  B )
9 simp3 959 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  =/=  .0.  )
10 eqid 2412 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
114, 10, 5opltn0 29685 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  ( lt
`  K ) X  <-> 
X  =/=  .0.  )
)
123, 8, 11syl2anc 643 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
(  .0.  ( lt
`  K ) X  <-> 
X  =/=  .0.  )
)
139, 12mpbird 224 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  .0.  ( lt `  K
) X )
14 atle.l . . . 4  |-  .<_  =  ( le `  K )
15 eqid 2412 . . . 4  |-  ( join `  K )  =  (
join `  K )
16 atle.a . . . 4  |-  A  =  ( Atoms `  K )
174, 14, 10, 15, 16hlrelat 29896 . . 3  |-  ( ( ( K  e.  HL  /\  .0.  e.  B  /\  X  e.  B )  /\  .0.  ( lt `  K ) X )  ->  E. p  e.  A  (  .0.  ( lt `  K ) (  .0.  ( join `  K
) p )  /\  (  .0.  ( join `  K
) p )  .<_  X ) )
181, 7, 8, 13, 17syl31anc 1187 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  (  .0.  ( lt `  K ) (  .0.  ( join `  K
) p )  /\  (  .0.  ( join `  K
) p )  .<_  X ) )
19 simpl1 960 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  K  e.  HL )
20 hlol 29856 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
2119, 20syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  K  e.  OL )
224, 16atbase 29784 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
2322adantl 453 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  p  e.  B )
244, 15, 5olj02 29721 . . . . . . 7  |-  ( ( K  e.  OL  /\  p  e.  B )  ->  (  .0.  ( join `  K ) p )  =  p )
2521, 23, 24syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  (  .0.  ( join `  K ) p )  =  p )
2625breq1d 4190 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  ( (  .0.  ( join `  K ) p )  .<_  X  <->  p  .<_  X ) )
2726biimpd 199 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  ( (  .0.  ( join `  K ) p )  .<_  X  ->  p 
.<_  X ) )
2827adantld 454 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  ( (  .0.  ( lt `  K ) (  .0.  ( join `  K
) p )  /\  (  .0.  ( join `  K
) p )  .<_  X )  ->  p  .<_  X ) )
2928reximdva 2786 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
( E. p  e.  A  (  .0.  ( lt `  K ) (  .0.  ( join `  K
) p )  /\  (  .0.  ( join `  K
) p )  .<_  X )  ->  E. p  e.  A  p  .<_  X ) )
3018, 29mpd 15 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  p  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   ltcplt 14361   joincjn 14364   0.cp0 14429   OPcops 29667   OLcol 29669   Atomscatm 29758   HLchlt 29845
This theorem is referenced by:  1cvratex  29967  llnle  30012  lhpexle  30499
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846
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