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Theorem atle 32653
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 981 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  K  e.  HL )
2 hlop 32580 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
323ad2ant1 1002 . . . 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 32402 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
73, 6syl 16 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  .0.  e.  B )
8 simp2 982 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  e.  B )
9 simp3 983 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  =/=  .0.  )
10 eqid 2433 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
114, 10, 5opltn0 32408 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  ( lt
`  K ) X  <-> 
X  =/=  .0.  )
)
123, 8, 11syl2anc 654 . . . 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 2433 . . . 4  |-  ( join `  K )  =  (
join `  K )
16 atle.a . . . 4  |-  A  =  ( Atoms `  K )
174, 14, 10, 15, 16hlrelat 32619 . . 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 1214 . 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 984 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  K  e.  HL )
20 hlol 32579 . . . . . . . 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 32507 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
2322adantl 463 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  p  e.  B )
244, 15, 5olj02 32444 . . . . . . 7  |-  ( ( K  e.  OL  /\  p  e.  B )  ->  (  .0.  ( join `  K ) p )  =  p )
2521, 23, 24syl2anc 654 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  /\  p  e.  A )  ->  (  .0.  ( join `  K ) p )  =  p )
2625breq1d 4290 . . . . 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 464 . . 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 2818 . 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596   E.wrex 2706   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   Basecbs 14157   lecple 14228   ltcplt 15094   joincjn 15097   0.cp0 15190   OPcops 32390   OLcol 32392   Atomscatm 32481   HLchlt 32568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-poset 15099  df-plt 15111  df-lub 15127  df-glb 15128  df-join 15129  df-meet 15130  df-p0 15192  df-lat 15199  df-clat 15261  df-oposet 32394  df-ol 32396  df-oml 32397  df-covers 32484  df-ats 32485  df-atl 32516  df-cvlat 32540  df-hlat 32569
This theorem is referenced by:  1cvratex  32690  llnle  32735  lhpexle  33222
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