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Theorem lanfwcl 14659
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 14657 . . . 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 2773 . . . . . 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 2773 . . . . 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 3875 . . . . . 6  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
98oveq2d 6089 . . . . 5  |-  ( x  =  A  ->  ( R  inf w  { x ,  y } )  =  ( R  inf w  { A ,  y } ) )
109eleq1d 2501 . . . 4  |-  ( x  =  A  ->  (
( R  inf w  { x ,  y } )  e.  X  <->  ( R  inf w  { A ,  y }
)  e.  X ) )
11 preq2 3876 . . . . . 6  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
1211oveq2d 6089 . . . . 5  |-  ( y  =  B  ->  ( R  inf w  { A ,  y } )  =  ( R  inf w  { A ,  B } ) )
1312eleq1d 2501 . . . 4  |-  ( y  =  B  ->  (
( R  inf w  { A ,  y } )  e.  X  <->  ( R  inf w  { A ,  B } )  e.  X
) )
1410, 13rspc2v 3050 . . 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 1652    e. wcel 1725   A.wral 2697   {cpr 3807   dom cdm 4870  (class class class)co 6073   PosetRelcps 14616    sup w cspw 14618    inf w cinf 14619   LatRelcla 14620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-dm 4880  df-iota 5410  df-fv 5454  df-ov 6076  df-lar 14625
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