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Theorem isla 14358
Description: The predicate "is a lattice" i.e. a poset in which any two elements have upper and lower bounds. (Contributed by NM, 12-Jun-2008.)
Hypothesis
Ref Expression
isla.1  |-  X  =  dom  R
Assertion
Ref Expression
isla  |-  ( 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
) ) )
Distinct variable groups:    x, y, R    x, X, y

Proof of Theorem isla
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 dmeq 4895 . . . 4  |-  ( r  =  R  ->  dom  r  =  dom  R )
2 isla.1 . . . 4  |-  X  =  dom  R
31, 2syl6eqr 2346 . . 3  |-  ( r  =  R  ->  dom  r  =  X )
4 oveq1 5881 . . . . . 6  |-  ( r  =  R  ->  (
r  sup w  { x ,  y } )  =  ( R  sup w  { x ,  y } ) )
54, 3eleq12d 2364 . . . . 5  |-  ( r  =  R  ->  (
( r  sup w  { x ,  y } )  e.  dom  r 
<->  ( R  sup w  { x ,  y } )  e.  X
) )
6 oveq1 5881 . . . . . 6  |-  ( r  =  R  ->  (
r  inf w  { x ,  y } )  =  ( R  inf w  { x ,  y } ) )
76, 3eleq12d 2364 . . . . 5  |-  ( r  =  R  ->  (
( r  inf w  { x ,  y } )  e.  dom  r 
<->  ( R  inf w  { x ,  y } )  e.  X
) )
85, 7anbi12d 691 . . . 4  |-  ( r  =  R  ->  (
( ( r  sup
w  { x ,  y } )  e. 
dom  r  /\  (
r  inf w  { x ,  y } )  e.  dom  r )  <-> 
( ( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X
) ) )
93, 8raleqbidv 2761 . . 3  |-  ( r  =  R  ->  ( A. y  e.  dom  r ( ( r  sup w  { x ,  y } )  e.  dom  r  /\  ( r  inf w  { x ,  y } )  e.  dom  r )  <->  A. y  e.  X  ( ( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X ) ) )
103, 9raleqbidv 2761 . 2  |-  ( r  =  R  ->  ( A. x  e.  dom  r A. y  e.  dom  r ( ( r  sup w  { x ,  y } )  e.  dom  r  /\  ( r  inf w  { x ,  y } )  e.  dom  r )  <->  A. x  e.  X  A. y  e.  X  ( ( R  sup w  { x ,  y } )  e.  X  /\  ( R  inf w  { x ,  y } )  e.  X ) ) )
11 df-lar 14326 . 2  |-  LatRel  =  {
r  e.  PosetRel  |  A. x  e.  dom  r A. y  e.  dom  r ( ( r  sup w  { x ,  y } )  e.  dom  r  /\  ( r  inf
w  { x ,  y } )  e. 
dom  r ) }
1210, 11elrab2 2938 1  |-  ( 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
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = 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 is referenced by:  laspwcl  14359  lanfwcl  14360  laps  14361  tolat  25389  toplat  25393  latdir  25398
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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