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Theorem clatlem 14216
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 14215 . . 3  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  A. s ( s  C_  B  ->  ( ( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) ) ) )
54simprbi 450 . 2  |-  ( K  e.  CLat  ->  A. s
( s  C_  B  ->  ( ( U `  s )  e.  B  /\  ( G `  s
)  e.  B ) ) )
6 fvex 5539 . . . . . 6  |-  ( Base `  K )  e.  _V
71, 6eqeltri 2353 . . . . 5  |-  B  e. 
_V
87elpw2 4175 . . . 4  |-  ( S  e.  ~P B  <->  S  C_  B
)
9 sseq1 3199 . . . . . 6  |-  ( s  =  S  ->  (
s  C_  B  <->  S  C_  B
) )
10 fveq2 5525 . . . . . . . 8  |-  ( s  =  S  ->  ( U `  s )  =  ( U `  S ) )
1110eleq1d 2349 . . . . . . 7  |-  ( s  =  S  ->  (
( U `  s
)  e.  B  <->  ( U `  S )  e.  B
) )
12 fveq2 5525 . . . . . . . 8  |-  ( s  =  S  ->  ( G `  s )  =  ( G `  S ) )
1312eleq1d 2349 . . . . . . 7  |-  ( s  =  S  ->  (
( G `  s
)  e.  B  <->  ( G `  S )  e.  B
) )
1411, 13anbi12d 691 . . . . . 6  |-  ( s  =  S  ->  (
( ( U `  s )  e.  B  /\  ( G `  s
)  e.  B )  <-> 
( ( U `  S )  e.  B  /\  ( G `  S
)  e.  B ) ) )
159, 14imbi12d 311 . . . . 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 2868 . . . 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 204 . . 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 45 . 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 455 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 358   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   ` cfv 5255   Basecbs 13148   Posetcpo 14074   lubclub 14076   glbcglb 14077   CLatccla 14213
This theorem is referenced by:  clatlubcl  14217  clatglbcl  14218
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  ax-sep 4141  ax-nul 4149
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-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-clat 14214
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