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Theorem poslubdg 14268
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 5543 . 2  |-  ( ph  ->  ( U `  S
)  =  ( ( lub `  K ) `
 S ) )
3 poslubdg.l . . 3  |-  .<_  =  ( le `  K )
4 eqid 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2296 . . 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 3227 . . 3  |-  ( ph  ->  S  C_  ( Base `  K ) )
10 poslubdg.t . . . 4  |-  ( ph  ->  T  e.  B )
1110, 8eleqtrd 2372 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
12 poslubdg.ub . . 3  |-  ( (
ph  /\  x  e.  S )  ->  x  .<_  T )
138eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  K
) ) )
1413biimpar 471 . . . . 5  |-  ( (
ph  /\  y  e.  ( Base `  K )
)  ->  y  e.  B )
15143adant3 975 . . . 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 1229 . . 3  |-  ( (
ph  /\  y  e.  ( Base `  K )  /\  A. x  e.  S  x  .<_  y )  ->  T  .<_  y )
183, 4, 5, 6, 9, 11, 12, 17poslubd 14267 . 2  |-  ( ph  ->  ( ( lub `  K
) `  S )  =  T )
192, 18eqtrd 2328 1  |-  ( ph  ->  ( U `  S
)  =  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   lubclub 14092
This theorem is referenced by:  posglbd  14269  mrelatlub  14305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-poset 14096  df-lub 14124
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