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Theorem meetat 29486
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 29403 . . . 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 29479 . . . 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 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
10 m.m . . . 4  |-  ./\  =  ( meet `  K )
115, 9, 10latmle2 14183 . . 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 29404 . . . 4  |-  ( K  e.  OL  ->  K  e.  OP )
14133ad2ant1 976 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  OP )
155, 10latmcl 14157 . . . 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 29482 . . 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   meetcmee 14079   0.cp0 14143   Latclat 14151   OPcops 29362   OLcol 29364   Atomscatm 29453
This theorem is referenced by:  meetat2  29487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-glb 14109  df-meet 14111  df-p0 14145  df-lat 14152  df-oposet 29366  df-ol 29368  df-covers 29456  df-ats 29457
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