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Theorem olm01 29971
Description: Meet with lattice zero is zero. (chm0 22985 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
olm0.b  |-  B  =  ( Base `  K
)
olm0.m  |-  ./\  =  ( meet `  K )
olm0.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
olm01  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .0.  )  =  .0.  )

Proof of Theorem olm01
StepHypRef Expression
1 olm0.b . 2  |-  B  =  ( Base `  K
)
2 eqid 2435 . 2  |-  ( le
`  K )  =  ( le `  K
)
3 ollat 29948 . . 3  |-  ( K  e.  OL  ->  K  e.  Lat )
43adantr 452 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  Lat )
5 simpr 448 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  X  e.  B )
6 olop 29949 . . . . 5  |-  ( K  e.  OL  ->  K  e.  OP )
76adantr 452 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  OP )
8 olm0.z . . . . 5  |-  .0.  =  ( 0. `  K )
91, 8op0cl 29919 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
107, 9syl 16 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  e.  B )
11 olm0.m . . . 4  |-  ./\  =  ( meet `  K )
121, 11latmcl 14472 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  ./\  .0.  )  e.  B )
134, 5, 10, 12syl3anc 1184 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .0.  )  e.  B )
141, 2, 11latmle2 14498 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  ./\  .0.  ) ( le `  K )  .0.  )
154, 5, 10, 14syl3anc 1184 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .0.  ) ( le `  K )  .0.  )
161, 2, 8op0le 29921 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
176, 16sylan 458 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
181, 2latref 14474 . . . 4  |-  ( ( K  e.  Lat  /\  .0.  e.  B )  ->  .0.  ( le `  K
)  .0.  )
194, 10, 18syl2anc 643 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K )  .0.  )
201, 2, 11latlem12 14499 . . . 4  |-  ( ( K  e.  Lat  /\  (  .0.  e.  B  /\  X  e.  B  /\  .0.  e.  B ) )  ->  ( (  .0.  ( le `  K
) X  /\  .0.  ( le `  K )  .0.  )  <->  .0.  ( le `  K ) ( X  ./\  .0.  )
) )
214, 10, 5, 10, 20syl13anc 1186 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( (  .0.  ( le `  K ) X  /\  .0.  ( le
`  K )  .0.  )  <->  .0.  ( le `  K ) ( X 
./\  .0.  ) )
)
2217, 19, 21mpbi2and 888 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K ) ( X 
./\  .0.  ) )
231, 2, 4, 13, 10, 15, 22latasymd 14478 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .0.  )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   meetcmee 14394   0.cp0 14458   Latclat 14466   OPcops 29907   OLcol 29909
This theorem is referenced by:  olm02  29972  omlfh1N  29993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-glb 14424  df-meet 14426  df-p0 14460  df-lat 14467  df-oposet 29911  df-ol 29913
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