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

Theorem lubl 15290
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 15288 . . . 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 463 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  ( U `  S )  .<_  z ) )
6 breq2 4296 . . . . . 6  |-  ( z  =  X  ->  (
y  .<_  z  <->  y  .<_  X ) )
76ralbidv 2735 . . . . 5  |-  ( z  =  X  ->  ( A. y  e.  S  y  .<_  z  <->  A. y  e.  S  y  .<_  X ) )
8 breq2 4296 . . . . 5  |-  ( z  =  X  ->  (
( U `  S
)  .<_  z  <->  ( U `  S )  .<_  X ) )
97, 8imbi12d 320 . . . 4  |-  ( z  =  X  ->  (
( A. y  e.  S  y  .<_  z  -> 
( U `  S
)  .<_  z )  <->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) ) )
109rspccva 3072 . . 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 471 . 2  |-  ( ( ( K  e.  CLat  /\  S  C_  B )  /\  X  e.  B
)  ->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) )
12113impa 1182 1  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  X  e.  B )  ->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715    C_ wss 3328   class class class wbr 4292   ` cfv 5418   Basecbs 14174   lecple 14245   lubclub 15112   CLatccla 15277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-lub 15144  df-clat 15278
This theorem is referenced by:  lubss  15291  lubun  15293
  Copyright terms: Public domain W3C validator