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Theorem poslubdg 14567
Description: Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypotheses
Ref Expression
poslubdg.l  |-  .<_  =  ( le `  K )
poslubdg.b  |-  ( ph  ->  B  =  ( Base `  K ) )
poslubdg.u  |-  ( ph  ->  U  =  ( lub `  K ) )
poslubdg.k  |-  ( ph  ->  K  e.  Poset )
poslubdg.s  |-  ( ph  ->  S  C_  B )
poslubdg.t  |-  ( ph  ->  T  e.  B )
poslubdg.ub  |-  ( (
ph  /\  x  e.  S )  ->  x  .<_  T )
poslubdg.le  |-  ( (
ph  /\  y  e.  B  /\  A. x  e.  S  x  .<_  y )  ->  T  .<_  y )
Assertion
Ref Expression
poslubdg  |-  ( ph  ->  ( U `  S
)  =  T )
Distinct variable groups:    x,  .<_ , y   
x, B, y    x, K, y    x, S, y   
x, U, y    x, T, y    ph, x, y

Proof of Theorem poslubdg
StepHypRef Expression
1 poslubdg.u . . 3  |-  ( ph  ->  U  =  ( lub `  K ) )
21fveq1d 5722 . 2  |-  ( ph  ->  ( U `  S
)  =  ( ( lub `  K ) `
 S ) )
3 poslubdg.l . . 3  |-  .<_  =  ( le `  K )
4 eqid 2435 . . 3  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2435 . . 3  |-  ( lub `  K )  =  ( lub `  K )
6 poslubdg.k . . 3  |-  ( ph  ->  K  e.  Poset )
7 poslubdg.s . . . 4  |-  ( ph  ->  S  C_  B )
8 poslubdg.b . . . 4  |-  ( ph  ->  B  =  ( Base `  K ) )
97, 8sseqtrd 3376 . . 3  |-  ( ph  ->  S  C_  ( Base `  K ) )
10 poslubdg.t . . . 4  |-  ( ph  ->  T  e.  B )
1110, 8eleqtrd 2511 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
12 poslubdg.ub . . 3  |-  ( (
ph  /\  x  e.  S )  ->  x  .<_  T )
138eleq2d 2502 . . . . . 6  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  K
) ) )
1413biimpar 472 . . . . 5  |-  ( (
ph  /\  y  e.  ( Base `  K )
)  ->  y  e.  B )
15143adant3 977 . . . 4  |-  ( (
ph  /\  y  e.  ( Base `  K )  /\  A. x  e.  S  x  .<_  y )  -> 
y  e.  B )
16 poslubdg.le . . . 4  |-  ( (
ph  /\  y  e.  B  /\  A. x  e.  S  x  .<_  y )  ->  T  .<_  y )
1715, 16syld3an2 1231 . . 3  |-  ( (
ph  /\  y  e.  ( Base `  K )  /\  A. x  e.  S  x  .<_  y )  ->  T  .<_  y )
183, 4, 5, 6, 9, 11, 12, 17poslubd 14566 . 2  |-  ( ph  ->  ( ( lub `  K
) `  S )  =  T )
192, 18eqtrd 2467 1  |-  ( ph  ->  ( U `  S
)  =  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Posetcpo 14389   lubclub 14391
This theorem is referenced by:  posglbd  14568  mrelatlub  14604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-riota 6541  df-poset 14395  df-lub 14423
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