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Theorem clatlem 14466
Description: Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
clatlem.b  |-  B  =  ( Base `  K
)
clatlem.u  |-  U  =  ( lub `  K
)
clatlem.g  |-  G  =  ( glb `  K
)
Assertion
Ref Expression
clatlem  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  (
( U `  S
)  e.  B  /\  ( G `  S )  e.  B ) )

Proof of Theorem clatlem
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 clatlem.b . . . 4  |-  B  =  ( Base `  K
)
2 clatlem.u . . . 4  |-  U  =  ( lub `  K
)
3 clatlem.g . . . 4  |-  G  =  ( glb `  K
)
41, 2, 3isclat 14465 . . 3  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  A. s ( s  C_  B  ->  ( ( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) ) ) )
54simprbi 451 . 2  |-  ( K  e.  CLat  ->  A. s
( s  C_  B  ->  ( ( U `  s )  e.  B  /\  ( G `  s
)  e.  B ) ) )
6 fvex 5682 . . . . . 6  |-  ( Base `  K )  e.  _V
71, 6eqeltri 2457 . . . . 5  |-  B  e. 
_V
87elpw2 4305 . . . 4  |-  ( S  e.  ~P B  <->  S  C_  B
)
9 sseq1 3312 . . . . . 6  |-  ( s  =  S  ->  (
s  C_  B  <->  S  C_  B
) )
10 fveq2 5668 . . . . . . . 8  |-  ( s  =  S  ->  ( U `  s )  =  ( U `  S ) )
1110eleq1d 2453 . . . . . . 7  |-  ( s  =  S  ->  (
( U `  s
)  e.  B  <->  ( U `  S )  e.  B
) )
12 fveq2 5668 . . . . . . . 8  |-  ( s  =  S  ->  ( G `  s )  =  ( G `  S ) )
1312eleq1d 2453 . . . . . . 7  |-  ( s  =  S  ->  (
( G `  s
)  e.  B  <->  ( G `  S )  e.  B
) )
1411, 13anbi12d 692 . . . . . 6  |-  ( s  =  S  ->  (
( ( U `  s )  e.  B  /\  ( G `  s
)  e.  B )  <-> 
( ( U `  S )  e.  B  /\  ( G `  S
)  e.  B ) ) )
159, 14imbi12d 312 . . . . 5  |-  ( s  =  S  ->  (
( s  C_  B  ->  ( ( U `  s )  e.  B  /\  ( G `  s
)  e.  B ) )  <->  ( S  C_  B  ->  ( ( U `
 S )  e.  B  /\  ( G `
 S )  e.  B ) ) ) )
1615spcgv 2979 . . . 4  |-  ( S  e.  ~P B  -> 
( A. s ( s  C_  B  ->  ( ( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) )  ->  ( S  C_  B  ->  ( ( U `
 S )  e.  B  /\  ( G `
 S )  e.  B ) ) ) )
178, 16sylbir 205 . . 3  |-  ( S 
C_  B  ->  ( A. s ( s  C_  B  ->  ( ( U `
 s )  e.  B  /\  ( G `
 s )  e.  B ) )  -> 
( S  C_  B  ->  ( ( U `  S )  e.  B  /\  ( G `  S
)  e.  B ) ) ) )
1817pm2.43a 47 . 2  |-  ( S 
C_  B  ->  ( A. s ( s  C_  B  ->  ( ( U `
 s )  e.  B  /\  ( G `
 s )  e.  B ) )  -> 
( ( U `  S )  e.  B  /\  ( G `  S
)  e.  B ) ) )
195, 18mpan9 456 1  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  (
( U `  S
)  e.  B  /\  ( G `  S )  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   ` cfv 5394   Basecbs 13396   Posetcpo 14324   lubclub 14326   glbcglb 14327   CLatccla 14463
This theorem is referenced by:  clatlubcl  14467  clatglbcl  14468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-clat 14464
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