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Theorem islat 14153
Description: The predicate "is a lattice." (Contributed by NM, 18-Oct-2012.)
Hypotheses
Ref Expression
islat.b  |-  B  =  ( Base `  K
)
islat.j  |-  .\/  =  ( join `  K )
islat.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
islat  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  A. x  e.  B  A. y  e.  B  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B ) ) )
Distinct variable groups:    x, y, B    x, K, y
Allowed substitution hints:    .\/ ( x, y)    ./\ (
x, y)

Proof of Theorem islat
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
2 islat.b . . . . 5  |-  B  =  ( Base `  K
)
31, 2syl6eqr 2333 . . . 4  |-  ( l  =  K  ->  ( Base `  l )  =  B )
43raleqdv 2742 . . . 4  |-  ( l  =  K  ->  ( A. y  e.  ( Base `  l ) ( ( x ( join `  l ) y )  e.  ( Base `  l
)  /\  ( x
( meet `  l )
y )  e.  (
Base `  l )
)  <->  A. y  e.  B  ( ( x (
join `  l )
y )  e.  (
Base `  l )  /\  ( x ( meet `  l ) y )  e.  ( Base `  l
) ) ) )
53, 4raleqbidv 2748 . . 3  |-  ( l  =  K  ->  ( A. x  e.  ( Base `  l ) A. y  e.  ( Base `  l ) ( ( x ( join `  l
) y )  e.  ( Base `  l
)  /\  ( x
( meet `  l )
y )  e.  (
Base `  l )
)  <->  A. x  e.  B  A. y  e.  B  ( ( x (
join `  l )
y )  e.  (
Base `  l )  /\  ( x ( meet `  l ) y )  e.  ( Base `  l
) ) ) )
6 fveq2 5525 . . . . . . . 8  |-  ( l  =  K  ->  ( join `  l )  =  ( join `  K
) )
7 islat.j . . . . . . . 8  |-  .\/  =  ( join `  K )
86, 7syl6eqr 2333 . . . . . . 7  |-  ( l  =  K  ->  ( join `  l )  = 
.\/  )
98oveqd 5875 . . . . . 6  |-  ( l  =  K  ->  (
x ( join `  l
) y )  =  ( x  .\/  y
) )
109, 3eleq12d 2351 . . . . 5  |-  ( l  =  K  ->  (
( x ( join `  l ) y )  e.  ( Base `  l
)  <->  ( x  .\/  y )  e.  B
) )
11 fveq2 5525 . . . . . . . 8  |-  ( l  =  K  ->  ( meet `  l )  =  ( meet `  K
) )
12 islat.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
1311, 12syl6eqr 2333 . . . . . . 7  |-  ( l  =  K  ->  ( meet `  l )  = 
./\  )
1413oveqd 5875 . . . . . 6  |-  ( l  =  K  ->  (
x ( meet `  l
) y )  =  ( x  ./\  y
) )
1514, 3eleq12d 2351 . . . . 5  |-  ( l  =  K  ->  (
( x ( meet `  l ) y )  e.  ( Base `  l
)  <->  ( x  ./\  y )  e.  B
) )
1610, 15anbi12d 691 . . . 4  |-  ( l  =  K  ->  (
( ( x (
join `  l )
y )  e.  (
Base `  l )  /\  ( x ( meet `  l ) y )  e.  ( Base `  l
) )  <->  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B ) ) )
17162ralbidv 2585 . . 3  |-  ( l  =  K  ->  ( A. x  e.  B  A. y  e.  B  ( ( x (
join `  l )
y )  e.  (
Base `  l )  /\  ( x ( meet `  l ) y )  e.  ( Base `  l
) )  <->  A. x  e.  B  A. y  e.  B  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B ) ) )
185, 17bitrd 244 . 2  |-  ( l  =  K  ->  ( A. x  e.  ( Base `  l ) A. y  e.  ( Base `  l ) ( ( x ( join `  l
) y )  e.  ( Base `  l
)  /\  ( x
( meet `  l )
y )  e.  (
Base `  l )
)  <->  A. x  e.  B  A. y  e.  B  ( ( x  .\/  y )  e.  B  /\  ( x  ./\  y
)  e.  B ) ) )
19 df-lat 14152 . 2  |-  Lat  =  { l  e.  Poset  | 
A. x  e.  (
Base `  l ) A. y  e.  ( Base `  l ) ( ( x ( join `  l ) y )  e.  ( Base `  l
)  /\  ( x
( meet `  l )
y )  e.  (
Base `  l )
) }
2018, 19elrab2 2925 1  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  A. x  e.  B  A. y  e.  B  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Posetcpo 14074   joincjn 14078   meetcmee 14079   Latclat 14151
This theorem is referenced by:  latlem  14154  latpos  14155  islati  14158  clatl  14220  odulatb  14247
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-iota 5219  df-fv 5263  df-ov 5861  df-lat 14152
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