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Theorem atle 29677
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 955 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  K  e.  HL )
2 hlop 29604 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
323ad2ant1 976 . . . 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 29426 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
73, 6syl 15 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  .0.  e.  B )
8 simp2 956 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  e.  B )
9 simp3 957 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  =/=  .0.  )
10 eqid 2358 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
114, 10, 5opltn0 29432 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  ( lt
`  K ) X  <-> 
X  =/=  .0.  )
)
123, 8, 11syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  -> 
(  .0.  ( lt
`  K ) X  <-> 
X  =/=  .0.  )
)
139, 12mpbird 223 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  .0.  ( lt `  K
) X )
14 atle.l . . . 4  |-  .<_  =  ( le `  K )
15 eqid 2358 . . . 4  |-  ( join `  K )  =  (
join `  K )
16 atle.a . . . 4  |-  A  =  ( Atoms `  K )
174, 14, 10, 15, 16hlrelat 29643 . . 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 1185 . 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 958 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  K  e.  HL )
20 hlol 29603 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
2119, 20syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  K  e.  OL )
224, 16atbase 29531 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
2322adantl 452 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  p  e.  B )
244, 15, 5olj02 29468 . . . . . . 7  |-  ( ( K  e.  OL  /\  p  e.  B )  ->  (  .0.  ( join `  K ) p )  =  p )
2521, 23, 24syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  (  .0.  ( join `  K ) p )  =  p )
2625breq1d 4112 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  ( (  .0.  ( join `  K ) p )  .<_  X  <->  p  .<_  X ) )
2726biimpd 198 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  ( (  .0.  ( join `  K ) p )  .<_  X  ->  p 
.<_  X ) )
2827adantld 453 . . 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 2731 . 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 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   Basecbs 13239   lecple 13306   ltcplt 14168   joincjn 14171   0.cp0 14236   OPcops 29414   OLcol 29416   Atomscatm 29505   HLchlt 29592
This theorem is referenced by:  1cvratex  29714  llnle  29759  lhpexle  30246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-lat 14245  df-clat 14307  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593
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