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Theorem lanfwcl 14587
Description: Closure of the infimum (meet) of two lattice elements. (Contributed by NM, 20-Jun-2008.)
Hypothesis
Ref Expression
isla.1  |-  X  =  dom  R
Assertion
Ref Expression
lanfwcl  |-  ( ( R  e.  LatRel  /\  A  e.  X  /\  B  e.  X )  ->  ( R  inf w  { A ,  B } )  e.  X )

Proof of Theorem lanfwcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isla.1 . . . . 5  |-  X  =  dom  R
21isla 14585 . . . 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 simpr 448 . . . . . . 7  |-  ( ( ( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X
)  ->  ( R  inf w  { x ,  y } )  e.  X )
43ralimi 2717 . . . . . 6  |-  ( A. y  e.  X  (
( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X
)  ->  A. y  e.  X  ( R  inf w  { x ,  y } )  e.  X )
54ralimi 2717 . . . . 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  inf w  { x ,  y } )  e.  X )
65adantl 453 . . . 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  inf w  { x ,  y } )  e.  X )
72, 6sylbi 188 . . 3  |-  ( R  e.  LatRel  ->  A. x  e.  X  A. y  e.  X  ( R  inf w  {
x ,  y } )  e.  X )
8 preq1 3819 . . . . . 6  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
98oveq2d 6029 . . . . 5  |-  ( x  =  A  ->  ( R  inf w  { x ,  y } )  =  ( R  inf w  { A ,  y } ) )
109eleq1d 2446 . . . 4  |-  ( x  =  A  ->  (
( R  inf w  { x ,  y } )  e.  X  <->  ( R  inf w  { A ,  y }
)  e.  X ) )
11 preq2 3820 . . . . . 6  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
1211oveq2d 6029 . . . . 5  |-  ( y  =  B  ->  ( R  inf w  { A ,  y } )  =  ( R  inf w  { A ,  B } ) )
1312eleq1d 2446 . . . 4  |-  ( y  =  B  ->  (
( R  inf w  { A ,  y } )  e.  X  <->  ( R  inf w  { A ,  B } )  e.  X
) )
1410, 13rspc2v 2994 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( R  inf w  { x ,  y } )  e.  X  ->  ( R  inf w  { A ,  B }
)  e.  X ) )
157, 14mpan9 456 . 2  |-  ( ( R  e.  LatRel  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( R  inf w  { A ,  B } )  e.  X
)
16153impb 1149 1  |-  ( ( R  e.  LatRel  /\  A  e.  X  /\  B  e.  X )  ->  ( R  inf w  { A ,  B } )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   {cpr 3751   dom cdm 4811  (class class class)co 6013   PosetRelcps 14544    sup w cspw 14546    inf w cinf 14547   LatRelcla 14548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-dm 4821  df-iota 5351  df-fv 5395  df-ov 6016  df-lar 14553
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