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Theorem clatlem 15955
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
StepHypRef Expression
1 clatlem.b . . 3  |-  B  =  ( Base `  K
)
2 clatlem.u . . 3  |-  U  =  ( lub `  K
)
3 simpl 455 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  K  e.  CLat )
4 fvex 5813 . . . . . . . 8  |-  ( Base `  K )  e.  _V
51, 4eqeltri 2484 . . . . . . 7  |-  B  e. 
_V
65elpw2 4555 . . . . . 6  |-  ( S  e.  ~P B  <->  S  C_  B
)
76biimpri 206 . . . . 5  |-  ( S 
C_  B  ->  S  e.  ~P B )
87adantl 464 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  ~P B )
9 clatlem.g . . . . . . . 8  |-  G  =  ( glb `  K
)
101, 2, 9isclat 15953 . . . . . . 7  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) ) )
1110biimpi 194 . . . . . 6  |-  ( K  e.  CLat  ->  ( K  e.  Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B
) ) )
1211adantr 463 . . . . 5  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( K  e.  Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) ) )
13 simpl 455 . . . . . 6  |-  ( ( dom  U  =  ~P B  /\  dom  G  =  ~P B )  ->  dom  U  =  ~P B
)
1413adantl 464 . . . . 5  |-  ( ( K  e.  Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) )  ->  dom  U  =  ~P B
)
1512, 14syl 17 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  dom  U  =  ~P B )
168, 15eleqtrrd 2491 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  U )
171, 2, 3, 16lubcl 15829 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )
1812simprrd 757 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  dom  G  =  ~P B )
198, 18eleqtrrd 2491 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  G )
201, 9, 3, 19glbcl 15842 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
2117, 20jca 530 1  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  (
( U `  S
)  e.  B  /\  ( G `  S )  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   _Vcvv 3056    C_ wss 3411   ~Pcpw 3952   dom cdm 4940   ` cfv 5523   Basecbs 14731   Posetcpo 15783   lubclub 15785   glbcglb 15786   CLatccla 15951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-lub 15818  df-glb 15819  df-clat 15952
This theorem is referenced by:  clatlubcl  15956  clatglbcl  15958
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