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Theorem meetat 30108
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 30025 . . . 4  |-  ( K  e.  OL  ->  K  e.  Lat )
213ad2ant1 976 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Lat )
3 simp2 956 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  X  e.  B )
4 simp3 957 . . . 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 30101 . . . 4  |-  ( P  e.  A  ->  P  e.  B )
84, 7syl 15 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  B )
9 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
10 m.m . . . 4  |-  ./\  =  ( meet `  K )
115, 9, 10latmle2 14199 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  ./\  P
) ( le `  K ) P )
122, 3, 8, 11syl3anc 1182 . 2  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  ( X  ./\  P
) ( le `  K ) P )
13 olop 30026 . . . 4  |-  ( K  e.  OL  ->  K  e.  OP )
14133ad2ant1 976 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  OP )
155, 10latmcl 14173 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  ./\  P
)  e.  B )
162, 3, 8, 15syl3anc 1182 . . 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 30104 . . 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 1182 . 2  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P ) ( le `  K ) P  <->  ( ( X  ./\  P )  =  P  \/  ( X 
./\  P )  =  .0.  ) ) )
2012, 19mpbid 201 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 176    \/ wo 357    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   meetcmee 14095   0.cp0 14159   Latclat 14167   OPcops 29984   OLcol 29986   Atomscatm 30075
This theorem is referenced by:  meetat2  30109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-glb 14125  df-meet 14127  df-p0 14161  df-lat 14168  df-oposet 29988  df-ol 29990  df-covers 30078  df-ats 30079
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