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Theorem latcl2 15535
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 5031 . . . 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 15534 . . . . 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 2558 . 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 2558 . 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 1379    e. wcel 1767   <.cop 4033    X. cxp 4997   dom cdm 4999   ` cfv 5588   Basecbs 14490   Posetcpo 15427   joincjn 15431   meetcmee 15432   Latclat 15532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-iota 5551  df-fv 5596  df-lat 15533
This theorem is referenced by:  latlej1  15547  latlej2  15548  latjle12  15549  latmle1  15563  latmle2  15564  latlem12  15565
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