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Theorem islat 14169
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 5541 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
2 islat.b . . . . 5  |-  B  =  ( Base `  K
)
31, 2syl6eqr 2346 . . . 4  |-  ( l  =  K  ->  ( Base `  l )  =  B )
43raleqdv 2755 . . . 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 2761 . . 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 5541 . . . . . . . 8  |-  ( l  =  K  ->  ( join `  l )  =  ( join `  K
) )
7 islat.j . . . . . . . 8  |-  .\/  =  ( join `  K )
86, 7syl6eqr 2346 . . . . . . 7  |-  ( l  =  K  ->  ( join `  l )  = 
.\/  )
98oveqd 5891 . . . . . 6  |-  ( l  =  K  ->  (
x ( join `  l
) y )  =  ( x  .\/  y
) )
109, 3eleq12d 2364 . . . . 5  |-  ( l  =  K  ->  (
( x ( join `  l ) y )  e.  ( Base `  l
)  <->  ( x  .\/  y )  e.  B
) )
11 fveq2 5541 . . . . . . . 8  |-  ( l  =  K  ->  ( meet `  l )  =  ( meet `  K
) )
12 islat.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
1311, 12syl6eqr 2346 . . . . . . 7  |-  ( l  =  K  ->  ( meet `  l )  = 
./\  )
1413oveqd 5891 . . . . . 6  |-  ( l  =  K  ->  (
x ( meet `  l
) y )  =  ( x  ./\  y
) )
1514, 3eleq12d 2364 . . . . 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 2598 . . 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 14168 . 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 2938 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 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   Posetcpo 14090   joincjn 14094   meetcmee 14095   Latclat 14167
This theorem is referenced by:  latlem  14170  latpos  14171  islati  14174  clatl  14236  odulatb  14263
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-iota 5235  df-fv 5279  df-ov 5877  df-lat 14168
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