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Theorem laspwcl 14668
Description: Closure of the supremum (join) of two lattice elements. (Contributed by NM, 12-Jun-2008.)
Hypothesis
Ref Expression
isla.1  |-  X  =  dom  R
Assertion
Ref Expression
laspwcl  |-  ( ( R  e.  LatRel  /\  A  e.  X  /\  B  e.  X )  ->  ( R  sup w  { A ,  B } )  e.  X )

Proof of Theorem laspwcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isla.1 . . . . 5  |-  X  =  dom  R
21isla 14667 . . . 4  |-  ( R  e.  LatRel 
<->  ( R  e.  PosetRel  /\  A. x  e.  X  A. y  e.  X  (
( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X
) ) )
3 simpl 445 . . . . . . 7  |-  ( ( ( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X
)  ->  ( R  sup w  { x ,  y } )  e.  X )
43ralimi 2783 . . . . . 6  |-  ( A. y  e.  X  (
( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X
)  ->  A. y  e.  X  ( R  sup w  { x ,  y } )  e.  X )
54ralimi 2783 . . . . 5  |-  ( A. x  e.  X  A. y  e.  X  (
( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X
)  ->  A. x  e.  X  A. y  e.  X  ( R  sup w  { x ,  y } )  e.  X )
65adantl 454 . . . 4  |-  ( ( R  e.  PosetRel  /\  A. x  e.  X  A. y  e.  X  (
( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X
) )  ->  A. x  e.  X  A. y  e.  X  ( R  sup w  { x ,  y } )  e.  X )
72, 6sylbi 189 . . 3  |-  ( R  e.  LatRel  ->  A. x  e.  X  A. y  e.  X  ( R  sup w  {
x ,  y } )  e.  X )
8 preq1 3885 . . . . . 6  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
98oveq2d 6099 . . . . 5  |-  ( x  =  A  ->  ( R  sup w  { x ,  y } )  =  ( R  sup w  { A ,  y } ) )
109eleq1d 2504 . . . 4  |-  ( x  =  A  ->  (
( R  sup w  { x ,  y } )  e.  X  <->  ( R  sup w  { A ,  y }
)  e.  X ) )
11 preq2 3886 . . . . . 6  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
1211oveq2d 6099 . . . . 5  |-  ( y  =  B  ->  ( R  sup w  { A ,  y } )  =  ( R  sup w  { A ,  B } ) )
1312eleq1d 2504 . . . 4  |-  ( y  =  B  ->  (
( R  sup w  { A ,  y } )  e.  X  <->  ( R  sup w  { A ,  B } )  e.  X
) )
1410, 13rspc2v 3060 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( R  sup w  { x ,  y } )  e.  X  ->  ( R  sup w  { A ,  B }
)  e.  X ) )
157, 14mpan9 457 . 2  |-  ( ( R  e.  LatRel  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( R  sup w  { A ,  B } )  e.  X
)
16153impb 1150 1  |-  ( ( R  e.  LatRel  /\  A  e.  X  /\  B  e.  X )  ->  ( R  sup w  { A ,  B } )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   {cpr 3817   dom cdm 4880  (class class class)co 6083   PosetRelcps 14626    sup w cspw 14628    inf w cinf 14629   LatRelcla 14630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-dm 4890  df-iota 5420  df-fv 5464  df-ov 6086  df-lar 14635
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