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Theorem olm01 30048
Description: Meet with lattice zero is zero. (chm0 22086 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 2296 . 2  |-  ( le
`  K )  =  ( le `  K
)
3 ollat 30025 . . 3  |-  ( K  e.  OL  ->  K  e.  Lat )
43adantr 451 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  Lat )
5 simpr 447 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  X  e.  B )
6 olop 30026 . . . . 5  |-  ( K  e.  OL  ->  K  e.  OP )
76adantr 451 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  OP )
8 olm0.z . . . . 5  |-  .0.  =  ( 0. `  K )
91, 8op0cl 29996 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
107, 9syl 15 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  e.  B )
11 olm0.m . . . 4  |-  ./\  =  ( meet `  K )
121, 11latmcl 14173 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  ./\  .0.  )  e.  B )
134, 5, 10, 12syl3anc 1182 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .0.  )  e.  B )
141, 2, 11latmle2 14199 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  ./\  .0.  ) ( le `  K )  .0.  )
154, 5, 10, 14syl3anc 1182 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .0.  ) ( le `  K )  .0.  )
161, 2, 8op0le 29998 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
176, 16sylan 457 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
181, 2latref 14175 . . . 4  |-  ( ( K  e.  Lat  /\  .0.  e.  B )  ->  .0.  ( le `  K
)  .0.  )
194, 10, 18syl2anc 642 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K )  .0.  )
201, 2, 11latlem12 14200 . . . 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 1184 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( (  .0.  ( le `  K ) X  /\  .0.  ( le
`  K )  .0.  )  <->  .0.  ( le `  K ) ( X 
./\  .0.  ) )
)
2217, 19, 21mpbi2and 887 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K ) ( X 
./\  .0.  ) )
231, 2, 4, 13, 10, 15, 22latasymd 14179 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .0.  )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = 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
This theorem is referenced by:  olm02  30049  omlfh1N  30070
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-glb 14125  df-meet 14127  df-p0 14161  df-lat 14168  df-oposet 29988  df-ol 29990
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