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Theorem laspwcl 14359
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 14358 . . . 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 443 . . . . . . 7  |-  ( ( ( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X
)  ->  ( R  sup w  { x ,  y } )  e.  X )
43ralimi 2631 . . . . . 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 2631 . . . . 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 452 . . . 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 187 . . 3  |-  ( R  e.  LatRel  ->  A. x  e.  X  A. y  e.  X  ( R  sup w  {
x ,  y } )  e.  X )
8 preq1 3719 . . . . . 6  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
98oveq2d 5890 . . . . 5  |-  ( x  =  A  ->  ( R  sup w  { x ,  y } )  =  ( R  sup w  { A ,  y } ) )
109eleq1d 2362 . . . 4  |-  ( x  =  A  ->  (
( R  sup w  { x ,  y } )  e.  X  <->  ( R  sup w  { A ,  y }
)  e.  X ) )
11 preq2 3720 . . . . . 6  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
1211oveq2d 5890 . . . . 5  |-  ( y  =  B  ->  ( R  sup w  { A ,  y } )  =  ( R  sup w  { A ,  B } ) )
1312eleq1d 2362 . . . 4  |-  ( y  =  B  ->  (
( R  sup w  { A ,  y } )  e.  X  <->  ( R  sup w  { A ,  B } )  e.  X
) )
1410, 13rspc2v 2903 . . 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 455 . 2  |-  ( ( R  e.  LatRel  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( R  sup w  { A ,  B } )  e.  X
)
16153impb 1147 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {cpr 3654   dom cdm 4705  (class class class)co 5874   PosetRelcps 14317    sup w cspw 14319    inf w cinf 14320   LatRelcla 14321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-dm 4715  df-iota 5235  df-fv 5279  df-ov 5877  df-lar 14326
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