MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clatlem Structured version   Unicode version

Theorem clatlem 15403
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 457 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  K  e.  CLat )
4 fvex 5812 . . . . . . . 8  |-  ( Base `  K )  e.  _V
51, 4eqeltri 2538 . . . . . . 7  |-  B  e. 
_V
65elpw2 4567 . . . . . 6  |-  ( S  e.  ~P B  <->  S  C_  B
)
76biimpri 206 . . . . 5  |-  ( S 
C_  B  ->  S  e.  ~P B )
87adantl 466 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  ~P B )
9 clatlem.g . . . . . . . 8  |-  G  =  ( glb `  K
)
101, 2, 9isclat 15401 . . . . . . 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 465 . . . . 5  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( K  e.  Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) ) )
13 simpl 457 . . . . . 6  |-  ( ( dom  U  =  ~P B  /\  dom  G  =  ~P B )  ->  dom  U  =  ~P B
)
1413adantl 466 . . . . 5  |-  ( ( K  e.  Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) )  ->  dom  U  =  ~P B
)
1512, 14syl 16 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  dom  U  =  ~P B )
168, 15eleqtrrd 2545 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  U )
171, 2, 3, 16lubcl 15277 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )
18 simprr 756 . . . . 5  |-  ( ( K  e.  Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) )  ->  dom  G  =  ~P B
)
1912, 18syl 16 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  dom  G  =  ~P B )
208, 19eleqtrrd 2545 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  G )
211, 9, 3, 20glbcl 15290 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
2217, 21jca 532 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439   ~Pcpw 3971   dom cdm 4951   ` cfv 5529   Basecbs 14295   Posetcpo 15232   lubclub 15234   glbcglb 15235   CLatccla 15399
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-lub 15266  df-glb 15267  df-clat 15400
This theorem is referenced by:  clatlubcl  15404  clatglbcl  15406
  Copyright terms: Public domain W3C validator