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Theorem latleeqm1 15826
Description: Less-than-or-equal-to in terms of meet. (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 15800 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  .<_  X )
433adant3 1014 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
54biantrurd 506 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<_  X  /\  X  .<_  Y ) ) )
6 simp1 994 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
7 simp2 995 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 simp3 996 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
9 latmle.m . . . . . 6  |-  ./\  =  ( meet `  K )
101, 2, 9latlem12 15825 . . . . 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 1228 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  X  /\  X  .<_  Y )  <-> 
X  .<_  ( X  ./\  Y ) ) )
125, 11bitrd 253 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  X  .<_  ( X  ./\  Y )
) )
131, 2, 9latmle1 15823 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
1413biantrurd 506 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  ( X 
./\  Y )  <->  ( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) ) ) )
1512, 14bitrd 253 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) ) ) )
16 latpos 15797 . . . 4  |-  ( K  e.  Lat  ->  K  e.  Poset )
17163ad2ant1 1015 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
181, 9latmcl 15799 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
191, 2posasymb 15699 . . 3  |-  ( ( K  e.  Poset  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X  ./\  Y ) )  <->  ( X  ./\ 
Y )  =  X ) )
2017, 18, 7, 19syl3anc 1226 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( X 
./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) )  <-> 
( X  ./\  Y
)  =  X ) )
2115, 20bitrd 253 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   Basecbs 14634   lecple 14709   Posetcpo 15686   meetcmee 15691   Latclat 15792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-preset 15674  df-poset 15692  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-lat 15793
This theorem is referenced by:  latleeqm2  15827  latnlemlt  15831  latabs2  15835  atnle  35455  2llnmat  35661  llnmlplnN  35676  dalem25  35835  2lnat  35921  lhpm0atN  36166  lhpmatb  36168  cdleme1  36365  cdleme5  36378  cdleme20d  36451  cdleme22e  36483  cdleme22eALTN  36484  cdleme23b  36489  cdleme32e  36584  doca2N  37266  djajN  37277  dihglblem5aN  37432  dihmeetbclemN  37444
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