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Theorem clatlubcl 14542
Description: LUB always exists in a complete lattice. (chsupcl 22844 analog.) (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
clatlubcl.b  |-  B  =  ( Base `  K
)
clatlubcl.u  |-  U  =  ( lub `  K
)
Assertion
Ref Expression
clatlubcl  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )

Proof of Theorem clatlubcl
StepHypRef Expression
1 clatlubcl.b . . 3  |-  B  =  ( Base `  K
)
2 clatlubcl.u . . 3  |-  U  =  ( lub `  K
)
3 eqid 2438 . . 3  |-  ( glb `  K )  =  ( glb `  K )
41, 2, 3clatlem 14541 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  (
( U `  S
)  e.  B  /\  ( ( glb `  K
) `  S )  e.  B ) )
54simpld 447 1  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   ` cfv 5456   Basecbs 13471   lubclub 14401   glbcglb 14402   CLatccla 14538
This theorem is referenced by:  lublem  14547  lubss  14550  lubun  14552  oduclatb  14573  clatp1ex  24196  lubunNEW  29833  atlatmstc  30179  polsubN  30766  2polvalN  30773  2polssN  30774  3polN  30775  2pmaplubN  30785  paddunN  30786  poldmj1N  30787  pnonsingN  30792  ispsubcl2N  30806  psubclinN  30807  paddatclN  30808  polsubclN  30811  poml4N  30812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-clat 14539
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