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Theorem latlem 15881
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 994 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
4 simp2 995 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
5 simp3 996 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
6 opelxpi 5020 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
763adant1 1012 . . . 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 15879 . . . . . 6  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) ) )
10 simprl 754 . . . . . 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 1015 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  dom  .\/  =  ( B  X.  B ) )
137, 12eleqtrrd 2545 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e. 
dom  .\/  )
141, 2, 3, 4, 5, 13joincl 15838 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
15 simprr 755 . . . . . 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 1015 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  dom  ./\  =  ( B  X.  B ) )
187, 17eleqtrrd 2545 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e. 
dom  ./\  )
191, 8, 3, 4, 5, 18meetcl 15852 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
2014, 19jca 530 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   <.cop 4022    X. cxp 4986   dom cdm 4988   ` cfv 5570  (class class class)co 6270   Basecbs 14719   Posetcpo 15771   joincjn 15775   meetcmee 15776   Latclat 15877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-lat 15878
This theorem is referenced by:  latjcl  15883  latmcl  15884
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