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Theorem clatglbcl2 15396
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 5802 . . . . . 6  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2535 . . . . 5  |-  B  e. 
_V
43elpw2 4557 . . . 4  |-  ( S  e.  ~P B  <->  S  C_  B
)
54biimpri 206 . . 3  |-  ( S 
C_  B  ->  S  e.  ~P B )
65adantl 466 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  ~P B )
7 eqid 2451 . . . . 5  |-  ( lub `  K )  =  ( lub `  K )
8 clatglbcl.g . . . . 5  |-  G  =  ( glb `  K
)
91, 7, 8isclat 15390 . . . 4  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  ( lub `  K )  =  ~P B  /\  dom  G  =  ~P B ) ) )
10 simprr 756 . . . 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 465 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  dom  G  =  ~P B )
136, 12eleqtrrd 2542 1  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071    C_ wss 3429   ~Pcpw 3961   dom cdm 4941   ` cfv 5519   Basecbs 14285   Posetcpo 15221   lubclub 15223   glbcglb 15224   CLatccla 15388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-dm 4951  df-iota 5482  df-fv 5527  df-clat 15389
This theorem is referenced by:  isglbd  15398  clatglb  15405  clatglble  15406
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