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Theorem poslubdg 15906
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 5874 . 2  |-  ( ph  ->  ( U `  S
)  =  ( ( lub `  K ) `
 S ) )
3 poslubdg.l . . 3  |-  .<_  =  ( le `  K )
4 eqid 2457 . . 3  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2457 . . 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 3535 . . 3  |-  ( ph  ->  S  C_  ( Base `  K ) )
10 poslubdg.t . . . 4  |-  ( ph  ->  T  e.  B )
1110, 8eleqtrd 2547 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
12 poslubdg.ub . . 3  |-  ( (
ph  /\  x  e.  S )  ->  x  .<_  T )
138eleq2d 2527 . . . . . 6  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  K
) ) )
1413biimpar 485 . . . . 5  |-  ( (
ph  /\  y  e.  ( Base `  K )
)  ->  y  e.  B )
15143adant3 1016 . . . 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 1275 . . 3  |-  ( (
ph  /\  y  e.  ( Base `  K )  /\  A. x  e.  S  x  .<_  y )  ->  T  .<_  y )
183, 4, 5, 6, 9, 11, 12, 17poslubd 15905 . 2  |-  ( ph  ->  ( ( lub `  K
) `  S )  =  T )
192, 18eqtrd 2498 1  |-  ( ph  ->  ( U `  S
)  =  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   class class class wbr 4456   ` cfv 5594   Basecbs 14644   lecple 14719   Posetcpo 15696   lubclub 15698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-preset 15684  df-poset 15702  df-lub 15731
This theorem is referenced by:  posglbd  15907  mrelatlub  15943
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