MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  latleeqm1 Unicode version

Theorem latleeqm1 14395
Description: Less-than-or-equal-to in terms of meet. (df-ss 3252 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 14369 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  .<_  X )
433adant3 976 . . . . 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 956 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
7 simp2 957 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 simp3 958 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
9 latmle.m . . . . . 6  |-  ./\  =  ( meet `  K )
101, 2, 9latlem12 14394 . . . . 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 1185 . . . 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 14392 . . . 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 14365 . . . 4  |-  ( K  e.  Lat  ->  K  e.  Poset )
17163ad2ant1 977 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
181, 9latmcl 14367 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
191, 2posasymb 14296 . . 3  |-  ( ( K  e.  Poset  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X  ./\  Y ) )  <->  ( X  ./\ 
Y )  =  X ) )
2017, 18, 7, 19syl3anc 1183 . 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 935    = wceq 1647    e. wcel 1715   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   Posetcpo 14284   meetcmee 14289   Latclat 14361
This theorem is referenced by:  latleeqm2  14396  latnlemlt  14400  latabs2  14404  atnle  29578  2llnmat  29784  llnmlplnN  29799  dalem25  29958  2lnat  30044  lhpm0atN  30289  lhpmatb  30291  cdleme1  30487  cdleme5  30500  cdleme20d  30572  cdleme22e  30604  cdleme22eALTN  30605  cdleme23b  30610  cdleme32e  30705  doca2N  31387  djajN  31398  dihglblem5aN  31553  dihmeetbclemN  31565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-glb 14319  df-meet 14321  df-lat 14362
  Copyright terms: Public domain W3C validator