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Theorem clatlem 14531
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 14530 . . 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 5734 . . . . . 6  |-  ( Base `  K )  e.  _V
71, 6eqeltri 2505 . . . . 5  |-  B  e. 
_V
87elpw2 4356 . . . 4  |-  ( S  e.  ~P B  <->  S  C_  B
)
9 sseq1 3361 . . . . . 6  |-  ( s  =  S  ->  (
s  C_  B  <->  S  C_  B
) )
10 fveq2 5720 . . . . . . . 8  |-  ( s  =  S  ->  ( U `  s )  =  ( U `  S ) )
1110eleq1d 2501 . . . . . . 7  |-  ( s  =  S  ->  (
( U `  s
)  e.  B  <->  ( U `  S )  e.  B
) )
12 fveq2 5720 . . . . . . . 8  |-  ( s  =  S  ->  ( G `  s )  =  ( G `  S ) )
1312eleq1d 2501 . . . . . . 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 3028 . . . 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 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   ` cfv 5446   Basecbs 13461   Posetcpo 14389   lubclub 14391   glbcglb 14392   CLatccla 14528
This theorem is referenced by:  clatlubcl  14532  clatglbcl  14533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-clat 14529
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