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Theorem latcl2 16245
Description: The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
latcl2.b  |-  B  =  ( Base `  K
)
latcl2.j  |-  .\/  =  ( join `  K )
latcl2.m  |-  ./\  =  ( meet `  K )
latcl2.k  |-  ( ph  ->  K  e.  Lat )
latcl2.x  |-  ( ph  ->  X  e.  B )
latcl2.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
latcl2  |-  ( ph  ->  ( <. X ,  Y >.  e.  dom  .\/  /\  <. X ,  Y >.  e. 
dom  ./\  ) )

Proof of Theorem latcl2
StepHypRef Expression
1 latcl2.x . . . 4  |-  ( ph  ->  X  e.  B )
2 latcl2.y . . . 4  |-  ( ph  ->  Y  e.  B )
3 opelxpi 4886 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
41, 2, 3syl2anc 665 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
5 latcl2.k . . . . 5  |-  ( ph  ->  K  e.  Lat )
6 latcl2.b . . . . . 6  |-  B  =  ( Base `  K
)
7 latcl2.j . . . . . 6  |-  .\/  =  ( join `  K )
8 latcl2.m . . . . . 6  |-  ./\  =  ( meet `  K )
96, 7, 8islat 16244 . . . . 5  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) ) )
105, 9sylib 199 . . . 4  |-  ( ph  ->  ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B
)  /\  dom  ./\  =  ( B  X.  B
) ) ) )
11 simprl 762 . . . 4  |-  ( ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) )  ->  dom  .\/  =  ( B  X.  B
) )
1210, 11syl 17 . . 3  |-  ( ph  ->  dom  .\/  =  ( B  X.  B ) )
134, 12eleqtrrd 2520 . 2  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  .\/  )
1410simprrd 765 . . 3  |-  ( ph  ->  dom  ./\  =  ( B  X.  B ) )
154, 14eleqtrrd 2520 . 2  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  ./\  )
1613, 15jca 534 1  |-  ( ph  ->  ( <. X ,  Y >.  e.  dom  .\/  /\  <. X ,  Y >.  e. 
dom  ./\  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   <.cop 4008    X. cxp 4852   dom cdm 4854   ` cfv 5601   Basecbs 15084   Posetcpo 16136   joincjn 16140   meetcmee 16141   Latclat 16242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-xp 4860  df-dm 4864  df-iota 5565  df-fv 5609  df-lat 16243
This theorem is referenced by:  latlej1  16257  latlej2  16258  latjle12  16259  latmle1  16273  latmle2  16274  latlem12  16275
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