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Theorem lubl 14224
Description: The LUB of a complete lattice subset is a 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 14222 . . . 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 449 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  ( U `  S )  .<_  z ) )
6 breq2 4027 . . . . . 6  |-  ( z  =  X  ->  (
y  .<_  z  <->  y  .<_  X ) )
76ralbidv 2563 . . . . 5  |-  ( z  =  X  ->  ( A. y  e.  S  y  .<_  z  <->  A. y  e.  S  y  .<_  X ) )
8 breq2 4027 . . . . 5  |-  ( z  =  X  ->  (
( U `  S
)  .<_  z  <->  ( U `  S )  .<_  X ) )
97, 8imbi12d 311 . . . 4  |-  ( z  =  X  ->  (
( A. y  e.  S  y  .<_  z  -> 
( U `  S
)  .<_  z )  <->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) ) )
109rspccva 2883 . . 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 457 . 2  |-  ( ( ( K  e.  CLat  /\  S  C_  B )  /\  X  e.  B
)  ->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S )  .<_  X ) )
12113impa 1146 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   lubclub 14076   CLatccla 14213
This theorem is referenced by:  lubss  14225  lubun  14227  lubunNEW  29163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-undef 6298  df-riota 6304  df-lub 14108  df-clat 14214
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