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Theorem atle 34107
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 991 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  K  e.  HL )
2 hlop 34034 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
323ad2ant1 1012 . . . 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 33856 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
73, 6syl 16 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  .0.  e.  B )
8 simp2 992 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  e.  B )
9 simp3 993 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  =/=  .0.  )
10 eqid 2460 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
114, 10, 5opltn0 33862 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  ( lt
`  K ) X  <-> 
X  =/=  .0.  )
)
123, 8, 11syl2anc 661 . . . 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 2460 . . . 4  |-  ( join `  K )  =  (
join `  K )
16 atle.a . . . 4  |-  A  =  ( Atoms `  K )
174, 14, 10, 15, 16hlrelat 34073 . . 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 1226 . 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 994 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  K  e.  HL )
20 hlol 34033 . . . . . . . 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 33961 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
2322adantl 466 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  p  e.  B )
244, 15, 5olj02 33898 . . . . . . 7  |-  ( ( K  e.  OL  /\  p  e.  B )  ->  (  .0.  ( join `  K ) p )  =  p )
2521, 23, 24syl2anc 661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  (  .0.  ( join `  K ) p )  =  p )
2625breq1d 4450 . . . . 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 467 . . 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 2931 . 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   ltcplt 15417   joincjn 15420   0.cp0 15513   OPcops 33844   OLcol 33846   Atomscatm 33935   HLchlt 34022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023
This theorem is referenced by:  1cvratex  34144  llnle  34189  lhpexle  34676
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