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Theorem latlem 15333
Description: Lemma for lattice properties. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
latlem.b  |-  B  =  ( Base `  K
)
latlem.j  |-  .\/  =  ( join `  K )
latlem.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latlem  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .\/  Y )  e.  B  /\  ( X  ./\  Y )  e.  B ) )

Proof of Theorem latlem
StepHypRef Expression
1 latlem.b . . 3  |-  B  =  ( Base `  K
)
2 latlem.j . . 3  |-  .\/  =  ( join `  K )
3 simp1 988 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
4 simp2 989 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
5 simp3 990 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
6 opelxpi 4974 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
763adant1 1006 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
8 latlem.m . . . . . . 7  |-  ./\  =  ( meet `  K )
91, 2, 8islat 15331 . . . . . 6  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) ) )
10 simprl 755 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) )  ->  dom  .\/  =  ( B  X.  B
) )
119, 10sylbi 195 . . . . 5  |-  ( K  e.  Lat  ->  dom  .\/  =  ( B  X.  B ) )
12113ad2ant1 1009 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  dom  .\/  =  ( B  X.  B ) )
137, 12eleqtrrd 2543 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e. 
dom  .\/  )
141, 2, 3, 4, 5, 13joincl 15290 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
15 simprr 756 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) )  ->  dom  ./\  =  ( B  X.  B
) )
169, 15sylbi 195 . . . . 5  |-  ( K  e.  Lat  ->  dom  ./\  =  ( B  X.  B ) )
17163ad2ant1 1009 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  dom  ./\  =  ( B  X.  B ) )
187, 17eleqtrrd 2543 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e. 
dom  ./\  )
191, 8, 3, 4, 5, 18meetcl 15304 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
2014, 19jca 532 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .\/  Y )  e.  B  /\  ( X  ./\  Y )  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   <.cop 3986    X. cxp 4941   dom cdm 4943   ` cfv 5521  (class class class)co 6195   Basecbs 14287   Posetcpo 15224   joincjn 15228   meetcmee 15229   Latclat 15329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-lat 15330
This theorem is referenced by:  latjcl  15335  latmcl  15336
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