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Theorem latleeqm1 14181
Description: Less-than-or-equal-to in terms of meet. (df-ss 3167 analog.) (Contributed by NM, 7-Nov-2011.)
Hypotheses
Ref Expression
latmle.b  |-  B  =  ( Base `  K
)
latmle.l  |-  .<_  =  ( le `  K )
latmle.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latleeqm1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )

Proof of Theorem latleeqm1
StepHypRef Expression
1 latmle.b . . . . . . 7  |-  B  =  ( Base `  K
)
2 latmle.l . . . . . . 7  |-  .<_  =  ( le `  K )
31, 2latref 14155 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  .<_  X )
433adant3 975 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
54biantrurd 494 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<_  X  /\  X  .<_  Y ) ) )
6 simp1 955 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
7 simp2 956 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 simp3 957 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
9 latmle.m . . . . . 6  |-  ./\  =  ( meet `  K )
101, 2, 9latlem12 14180 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( X  .<_  X  /\  X  .<_  Y )  <->  X  .<_  ( X  ./\  Y )
) )
116, 7, 7, 8, 10syl13anc 1184 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  X  /\  X  .<_  Y )  <-> 
X  .<_  ( X  ./\  Y ) ) )
125, 11bitrd 244 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  X  .<_  ( X  ./\  Y )
) )
131, 2, 9latmle1 14178 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
1413biantrurd 494 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  ( X 
./\  Y )  <->  ( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) ) ) )
1512, 14bitrd 244 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) ) ) )
16 latpos 14151 . . . 4  |-  ( K  e.  Lat  ->  K  e.  Poset )
17163ad2ant1 976 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
181, 9latmcl 14153 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
191, 2posasymb 14082 . . 3  |-  ( ( K  e.  Poset  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X  ./\  Y ) )  <->  ( X  ./\ 
Y )  =  X ) )
2017, 18, 7, 19syl3anc 1182 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( X 
./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) )  <-> 
( X  ./\  Y
)  =  X ) )
2115, 20bitrd 244 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   Posetcpo 14070   meetcmee 14075   Latclat 14147
This theorem is referenced by:  latleeqm2  14182  latnlemlt  14186  latabs2  14190  atnle  28786  2llnmat  28992  llnmlplnN  29007  dalem25  29166  2lnat  29252  lhpm0atN  29497  lhpmatb  29499  cdleme1  29695  cdleme5  29708  cdleme20d  29780  cdleme22e  29812  cdleme22eALTN  29813  cdleme23b  29818  cdleme32e  29913  doca2N  30595  djajN  30606  dihglblem5aN  30761  dihmeetbclemN  30773
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14076  df-glb 14105  df-meet 14107  df-lat 14148
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