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Theorem atle 32453
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 997 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  K  e.  HL )
2 hlop 32380 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
323ad2ant1 1018 . . . 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 32202 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
73, 6syl 17 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  .0.  e.  B )
8 simp2 998 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  e.  B )
9 simp3 999 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  =/=  .0.  )
10 eqid 2402 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
114, 10, 5opltn0 32208 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  ( lt
`  K ) X  <-> 
X  =/=  .0.  )
)
123, 8, 11syl2anc 659 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
(  .0.  ( lt
`  K ) X  <-> 
X  =/=  .0.  )
)
139, 12mpbird 232 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  .0.  ( lt `  K
) X )
14 atle.l . . . 4  |-  .<_  =  ( le `  K )
15 eqid 2402 . . . 4  |-  ( join `  K )  =  (
join `  K )
16 atle.a . . . 4  |-  A  =  ( Atoms `  K )
174, 14, 10, 15, 16hlrelat 32419 . . 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 1233 . 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 1000 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  K  e.  HL )
20 hlol 32379 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
2119, 20syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  K  e.  OL )
224, 16atbase 32307 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
2322adantl 464 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  p  e.  B )
244, 15, 5olj02 32244 . . . . . . 7  |-  ( ( K  e.  OL  /\  p  e.  B )  ->  (  .0.  ( join `  K ) p )  =  p )
2521, 23, 24syl2anc 659 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  (  .0.  ( join `  K ) p )  =  p )
2625breq1d 4405 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  ( (  .0.  ( join `  K ) p )  .<_  X  <->  p  .<_  X ) )
2726biimpd 207 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  ( (  .0.  ( join `  K ) p )  .<_  X  ->  p 
.<_  X ) )
2827adantld 465 . . 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 2879 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   ltcplt 15894   joincjn 15897   0.cp0 15991   OPcops 32190   OLcol 32192   Atomscatm 32281   HLchlt 32368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369
This theorem is referenced by:  1cvratex  32490  llnle  32535  lhpexle  33022
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