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Theorem lubl 14474
Description: The LUB of a complete lattice subset is the least bound. (Contributed by NM, 19-Oct-2011.)
Hypotheses
Ref Expression
lublem.b  |-  B  =  ( Base `  K
)
lublem.l  |-  .<_  =  ( le `  K )
lublem.u  |-  U  =  ( lub `  K
)
Assertion
Ref Expression
lubl  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  X  e.  B )  ->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) )
Distinct variable groups:    y, K    y, S    y, U    y,  .<_   
y, X
Allowed substitution hint:    B( y)

Proof of Theorem lubl
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 lublem.b . . . . 5  |-  B  =  ( Base `  K
)
2 lublem.l . . . . 5  |-  .<_  =  ( le `  K )
3 lublem.u . . . . 5  |-  U  =  ( lub `  K
)
41, 2, 3lublem 14472 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( A. y  e.  S  y  .<_  ( U `  S )  /\  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  ( U `  S )  .<_  z ) ) )
54simprd 450 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  ( U `  S )  .<_  z ) )
6 breq2 4157 . . . . . 6  |-  ( z  =  X  ->  (
y  .<_  z  <->  y  .<_  X ) )
76ralbidv 2669 . . . . 5  |-  ( z  =  X  ->  ( A. y  e.  S  y  .<_  z  <->  A. y  e.  S  y  .<_  X ) )
8 breq2 4157 . . . . 5  |-  ( z  =  X  ->  (
( U `  S
)  .<_  z  <->  ( U `  S )  .<_  X ) )
97, 8imbi12d 312 . . . 4  |-  ( z  =  X  ->  (
( A. y  e.  S  y  .<_  z  -> 
( U `  S
)  .<_  z )  <->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) ) )
109rspccva 2994 . . 3  |-  ( ( A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  ( U `  S )  .<_  z )  /\  X  e.  B )  ->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) )
115, 10sylan 458 . 2  |-  ( ( ( K  e.  CLat  /\  S  C_  B )  /\  X  e.  B
)  ->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) )
12113impa 1148 1  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  X  e.  B )  ->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649    C_ wss 3263   class class class wbr 4153   ` cfv 5394   Basecbs 13396   lecple 13463   lubclub 14326   CLatccla 14463
This theorem is referenced by:  lubss  14475  lubun  14477  lubunNEW  29088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-undef 6479  df-riota 6485  df-lub 14358  df-clat 14464
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