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Theorem islat 14481
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 5731 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
2 islat.b . . . . 5  |-  B  =  ( Base `  K
)
31, 2syl6eqr 2488 . . . 4  |-  ( l  =  K  ->  ( Base `  l )  =  B )
43raleqdv 2912 . . . 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 2918 . . 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 5731 . . . . . . . 8  |-  ( l  =  K  ->  ( join `  l )  =  ( join `  K
) )
7 islat.j . . . . . . . 8  |-  .\/  =  ( join `  K )
86, 7syl6eqr 2488 . . . . . . 7  |-  ( l  =  K  ->  ( join `  l )  = 
.\/  )
98oveqd 6101 . . . . . 6  |-  ( l  =  K  ->  (
x ( join `  l
) y )  =  ( x  .\/  y
) )
109, 3eleq12d 2506 . . . . 5  |-  ( l  =  K  ->  (
( x ( join `  l ) y )  e.  ( Base `  l
)  <->  ( x  .\/  y )  e.  B
) )
11 fveq2 5731 . . . . . . . 8  |-  ( l  =  K  ->  ( meet `  l )  =  ( meet `  K
) )
12 islat.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
1311, 12syl6eqr 2488 . . . . . . 7  |-  ( l  =  K  ->  ( meet `  l )  = 
./\  )
1413oveqd 6101 . . . . . 6  |-  ( l  =  K  ->  (
x ( meet `  l
) y )  =  ( x  ./\  y
) )
1514, 3eleq12d 2506 . . . . 5  |-  ( l  =  K  ->  (
( x ( meet `  l ) y )  e.  ( Base `  l
)  <->  ( x  ./\  y )  e.  B
) )
1610, 15anbi12d 693 . . . 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 2749 . . 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 246 . 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 14480 . 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 3096 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5457  (class class class)co 6084   Basecbs 13474   Posetcpo 14402   joincjn 14406   meetcmee 14407   Latclat 14479
This theorem is referenced by:  latlem  14482  latpos  14483  islati  14486  clatl  14548  odulatb  14575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-lat 14480
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