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Theorem clatglbcl2 15944
Description: Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
Hypotheses
Ref Expression
clatglbcl.b  |-  B  =  ( Base `  K
)
clatglbcl.g  |-  G  =  ( glb `  K
)
Assertion
Ref Expression
clatglbcl2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  G )

Proof of Theorem clatglbcl2
StepHypRef Expression
1 clatglbcl.b . . . . . 6  |-  B  =  ( Base `  K
)
2 fvex 5858 . . . . . 6  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2538 . . . . 5  |-  B  e. 
_V
43elpw2 4601 . . . 4  |-  ( S  e.  ~P B  <->  S  C_  B
)
54biimpri 206 . . 3  |-  ( S 
C_  B  ->  S  e.  ~P B )
65adantl 464 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  ~P B )
7 eqid 2454 . . . . 5  |-  ( lub `  K )  =  ( lub `  K )
8 clatglbcl.g . . . . 5  |-  G  =  ( glb `  K
)
91, 7, 8isclat 15938 . . . 4  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  ( lub `  K )  =  ~P B  /\  dom  G  =  ~P B ) ) )
10 simprr 755 . . . 4  |-  ( ( K  e.  Poset  /\  ( dom  ( lub `  K
)  =  ~P B  /\  dom  G  =  ~P B ) )  ->  dom  G  =  ~P B
)
119, 10sylbi 195 . . 3  |-  ( K  e.  CLat  ->  dom  G  =  ~P B )
1211adantr 463 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  dom  G  =  ~P B )
136, 12eleqtrrd 2545 1  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   dom cdm 4988   ` cfv 5570   Basecbs 14716   Posetcpo 15768   lubclub 15770   glbcglb 15771   CLatccla 15936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-dm 4998  df-iota 5534  df-fv 5578  df-clat 15937
This theorem is referenced by:  isglbd  15946  clatglb  15953  clatglble  15954
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