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

Theorem poslubdg 14252
Description: Properties which determine a 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 5527 . 2  |-  ( ph  ->  ( U `  S
)  =  ( ( lub `  K ) `
 S ) )
3 poslubdg.l . . 3  |-  .<_  =  ( le `  K )
4 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2283 . . 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 3214 . . 3  |-  ( ph  ->  S  C_  ( Base `  K ) )
10 poslubdg.t . . . 4  |-  ( ph  ->  T  e.  B )
1110, 8eleqtrd 2359 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
12 poslubdg.ub . . 3  |-  ( (
ph  /\  x  e.  S )  ->  x  .<_  T )
138eleq2d 2350 . . . . . 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 14251 . 2  |-  ( ph  ->  ( ( lub `  K
) `  S )  =  T )
192, 18eqtrd 2315 1  |-  ( ph  ->  ( U `  S
)  =  T )
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   Posetcpo 14074   lubclub 14076
This theorem is referenced by:  posglbd  14253  mrelatlub  14289
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-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
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-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-riota 6304  df-poset 14080  df-lub 14108
  Copyright terms: Public domain W3C validator