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Theorem isla 14342
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 4879 . . . 4  |-  ( r  =  R  ->  dom  r  =  dom  R )
2 isla.1 . . . 4  |-  X  =  dom  R
31, 2syl6eqr 2333 . . 3  |-  ( r  =  R  ->  dom  r  =  X )
4 oveq1 5865 . . . . . 6  |-  ( r  =  R  ->  (
r  sup w  { x ,  y } )  =  ( R  sup w  { x ,  y } ) )
54, 3eleq12d 2351 . . . . 5  |-  ( r  =  R  ->  (
( r  sup w  { x ,  y } )  e.  dom  r 
<->  ( R  sup w  { x ,  y } )  e.  X
) )
6 oveq1 5865 . . . . . 6  |-  ( r  =  R  ->  (
r  inf w  { x ,  y } )  =  ( R  inf w  { x ,  y } ) )
76, 3eleq12d 2351 . . . . 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 2748 . . 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 2748 . 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 14310 . 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 2925 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 1623    e. wcel 1684   A.wral 2543   {cpr 3641   dom cdm 4689  (class class class)co 5858   PosetRelcps 14301    sup w cspw 14303    inf w cinf 14304   LatRelcla 14305
This theorem is referenced by:  laspwcl  14343  lanfwcl  14344  laps  14345  tolat  25286  toplat  25290  latdir  25295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861  df-lar 14310
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