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Theorem meetat 29779
Description: The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)
Hypotheses
Ref Expression
m.b  |-  B  =  ( Base `  K
)
m.m  |-  ./\  =  ( meet `  K )
m.z  |-  .0.  =  ( 0. `  K )
m.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
meetat  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P )  =  P  \/  ( X  ./\  P )  =  .0.  ) )

Proof of Theorem meetat
StepHypRef Expression
1 ollat 29696 . . . 4  |-  ( K  e.  OL  ->  K  e.  Lat )
213ad2ant1 978 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Lat )
3 simp2 958 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  X  e.  B )
4 simp3 959 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  A )
5 m.b . . . . 5  |-  B  =  ( Base `  K
)
6 m.a . . . . 5  |-  A  =  ( Atoms `  K )
75, 6atbase 29772 . . . 4  |-  ( P  e.  A  ->  P  e.  B )
84, 7syl 16 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  B )
9 eqid 2404 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
10 m.m . . . 4  |-  ./\  =  ( meet `  K )
115, 9, 10latmle2 14461 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  ./\  P
) ( le `  K ) P )
122, 3, 8, 11syl3anc 1184 . 2  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  ( X  ./\  P
) ( le `  K ) P )
13 olop 29697 . . . 4  |-  ( K  e.  OL  ->  K  e.  OP )
14133ad2ant1 978 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  OP )
155, 10latmcl 14435 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  ./\  P
)  e.  B )
162, 3, 8, 15syl3anc 1184 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  ( X  ./\  P
)  e.  B )
17 m.z . . . 4  |-  .0.  =  ( 0. `  K )
185, 9, 17, 6leatb 29775 . . 3  |-  ( ( K  e.  OP  /\  ( X  ./\  P )  e.  B  /\  P  e.  A )  ->  (
( X  ./\  P
) ( le `  K ) P  <->  ( ( X  ./\  P )  =  P  \/  ( X 
./\  P )  =  .0.  ) ) )
1914, 16, 4, 18syl3anc 1184 . 2  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P ) ( le `  K ) P  <->  ( ( X  ./\  P )  =  P  \/  ( X 
./\  P )  =  .0.  ) ) )
2012, 19mpbid 202 1  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P )  =  P  \/  ( X  ./\  P )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   meetcmee 14357   0.cp0 14421   Latclat 14429   OPcops 29655   OLcol 29657   Atomscatm 29746
This theorem is referenced by:  meetat2  29780
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-glb 14387  df-meet 14389  df-p0 14423  df-lat 14430  df-oposet 29659  df-ol 29661  df-covers 29749  df-ats 29750
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