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Theorem mrelatlub 15361
Description: Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypotheses
Ref Expression
mreclat.i  |-  I  =  (toInc `  C )
mrelatlub.f  |-  F  =  (mrCls `  C )
mrelatlub.l  |-  L  =  ( lub `  I
)
Assertion
Ref Expression
mrelatlub  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  ( L `  U )  =  ( F `  U. U ) )

Proof of Theorem mrelatlub
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . 2  |-  ( le
`  I )  =  ( le `  I
)
2 mreclat.i . . . 4  |-  I  =  (toInc `  C )
32ipobas 15330 . . 3  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
43adantr 465 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  C  =  ( Base `  I
) )
5 mrelatlub.l . . 3  |-  L  =  ( lub `  I
)
65a1i 11 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  L  =  ( lub `  I
) )
72ipopos 15335 . . 3  |-  I  e. 
Poset
87a1i 11 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  I  e.  Poset )
9 simpr 461 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U  C_  C )
10 uniss 4117 . . . . 5  |-  ( U 
C_  C  ->  U. U  C_ 
U. C )
1110adantl 466 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_ 
U. C )
12 mreuni 14543 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
1312adantr 465 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. C  =  X )
1411, 13sseqtrd 3397 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_  X )
15 mrelatlub.f . . . 4  |-  F  =  (mrCls `  C )
1615mrccl 14554 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  -> 
( F `  U. U )  e.  C
)
1714, 16syldan 470 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  ( F `  U. U )  e.  C )
18 elssuni 4126 . . . 4  |-  ( x  e.  U  ->  x  C_ 
U. U )
1915mrcssid 14560 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  ->  U. U  C_  ( F `
 U. U ) )
2014, 19syldan 470 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_  ( F `  U. U ) )
2118, 20sylan9ssr 3375 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  x  C_  ( F `  U. U ) )
22 simpll 753 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
239sselda 3361 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  x  e.  C )
2417adantr 465 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  ( F `  U. U )  e.  C )
252, 1ipole 15333 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  C  /\  ( F `  U. U )  e.  C )  -> 
( x ( le
`  I ) ( F `  U. U
)  <->  x  C_  ( F `
 U. U ) ) )
2622, 23, 24, 25syl3anc 1218 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  (
x ( le `  I ) ( F `
 U. U )  <-> 
x  C_  ( F `  U. U ) ) )
2721, 26mpbird 232 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  x
( le `  I
) ( F `  U. U ) )
28 simp1l 1012 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  C  e.  (Moore `  X ) )
29 simplll 757 . . . . . . . . 9  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
30 simplr 754 . . . . . . . . . 10  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C )  ->  U  C_  C )
3130sselda 3361 . . . . . . . . 9  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  x  e.  C )
32 simplr 754 . . . . . . . . 9  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  y  e.  C )
332, 1ipole 15333 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  C  /\  y  e.  C )  ->  (
x ( le `  I ) y  <->  x  C_  y
) )
3429, 31, 32, 33syl3anc 1218 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  (
x ( le `  I ) y  <->  x  C_  y
) )
3534biimpd 207 . . . . . . 7  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  (
x ( le `  I ) y  ->  x  C_  y ) )
3635ralimdva 2799 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C )  ->  ( A. x  e.  U  x ( le `  I ) y  ->  A. x  e.  U  x  C_  y ) )
37363impia 1184 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  A. x  e.  U  x  C_  y
)
38 unissb 4128 . . . . 5  |-  ( U. U  C_  y  <->  A. x  e.  U  x  C_  y
)
3937, 38sylibr 212 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  U. U  C_  y )
40 simp2 989 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  y  e.  C )
4115mrcsscl 14563 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  y  /\  y  e.  C )  ->  ( F `  U. U ) 
C_  y )
4228, 39, 40, 41syl3anc 1218 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  ( F `  U. U )  C_  y )
43173ad2ant1 1009 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  ( F `  U. U )  e.  C )
442, 1ipole 15333 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  ( F `  U. U )  e.  C  /\  y  e.  C )  ->  (
( F `  U. U ) ( le
`  I ) y  <-> 
( F `  U. U )  C_  y
) )
4528, 43, 40, 44syl3anc 1218 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  ( ( F `  U. U ) ( le `  I
) y  <->  ( F `  U. U )  C_  y ) )
4642, 45mpbird 232 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  ( F `  U. U ) ( le `  I ) y )
471, 4, 6, 8, 9, 17, 27, 46poslubdg 15324 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  ( L `  U )  =  ( F `  U. U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720    C_ wss 3333   U.cuni 4096   class class class wbr 4297   ` cfv 5423   Basecbs 14179   lecple 14250  Moorecmre 14525  mrClscmrc 14526   Posetcpo 15115   lubclub 15117  toInccipo 15326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-tset 14262  df-ple 14263  df-ocomp 14264  df-mre 14529  df-mrc 14530  df-poset 15121  df-lub 15149  df-ipo 15327
This theorem is referenced by:  mreclatBAD  15362
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