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Theorem latcl2 15329
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 4972 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
41, 2, 3syl2anc 661 . . 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 15328 . . . . 5  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) ) )
105, 9sylib 196 . . . 4  |-  ( ph  ->  ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B
)  /\  dom  ./\  =  ( B  X.  B
) ) ) )
11 simprl 755 . . . 4  |-  ( ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) )  ->  dom  .\/  =  ( B  X.  B
) )
1210, 11syl 16 . . 3  |-  ( ph  ->  dom  .\/  =  ( B  X.  B ) )
134, 12eleqtrrd 2542 . 2  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  .\/  )
14 simprr 756 . . . 4  |-  ( ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) )  ->  dom  ./\  =  ( B  X.  B
) )
1510, 14syl 16 . . 3  |-  ( ph  ->  dom  ./\  =  ( B  X.  B ) )
164, 15eleqtrrd 2542 . 2  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  ./\  )
1713, 16jca 532 1  |-  ( ph  ->  ( <. X ,  Y >.  e.  dom  .\/  /\  <. X ,  Y >.  e. 
dom  ./\  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3984    X. cxp 4939   dom cdm 4941   ` cfv 5519   Basecbs 14285   Posetcpo 15221   joincjn 15225   meetcmee 15226   Latclat 15326
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
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-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-xp 4947  df-dm 4951  df-iota 5482  df-fv 5527  df-lat 15327
This theorem is referenced by:  latlej1  15341  latlej2  15342  latjle12  15343  latmle1  15357  latmle2  15358  latlem12  15359
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