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Theorem mrelatlub 14604
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 2435 . 2  |-  ( le
`  I )  =  ( le `  I
)
2 mreclat.i . . . 4  |-  I  =  (toInc `  C )
32ipobas 14573 . . 3  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
43adantr 452 . 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 14578 . . 3  |-  I  e. 
Poset
87a1i 11 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  I  e.  Poset )
9 simpr 448 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U  C_  C )
10 uniss 4028 . . . . 5  |-  ( U 
C_  C  ->  U. U  C_ 
U. C )
1110adantl 453 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_ 
U. C )
12 mreuni 13817 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
1312adantr 452 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. C  =  X )
1411, 13sseqtrd 3376 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_  X )
15 mrelatlub.f . . . 4  |-  F  =  (mrCls `  C )
1615mrccl 13828 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  -> 
( F `  U. U )  e.  C
)
1714, 16syldan 457 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  ( F `  U. U )  e.  C )
18 elssuni 4035 . . . 4  |-  ( x  e.  U  ->  x  C_ 
U. U )
1915mrcssid 13834 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  ->  U. U  C_  ( F `
 U. U ) )
2014, 19syldan 457 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_  ( F `  U. U ) )
2118, 20sylan9ssr 3354 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  x  C_  ( F `  U. U ) )
22 simpll 731 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
239sselda 3340 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  x  e.  C )
2417adantr 452 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  ( F `  U. U )  e.  C )
252, 1ipole 14576 . . . 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 1184 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  (
x ( le `  I ) ( F `
 U. U )  <-> 
x  C_  ( F `  U. U ) ) )
2721, 26mpbird 224 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  x
( le `  I
) ( F `  U. U ) )
28 simp1l 981 . . . 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 735 . . . . . . . . 9  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
30 simplr 732 . . . . . . . . . 10  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C )  ->  U  C_  C )
3130sselda 3340 . . . . . . . . 9  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  x  e.  C )
32 simplr 732 . . . . . . . . 9  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  y  e.  C )
332, 1ipole 14576 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  C  /\  y  e.  C )  ->  (
x ( le `  I ) y  <->  x  C_  y
) )
3429, 31, 32, 33syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  (
x ( le `  I ) y  <->  x  C_  y
) )
3534biimpd 199 . . . . . . 7  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  (
x ( le `  I ) y  ->  x  C_  y ) )
3635ralimdva 2776 . . . . . 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 1150 . . . . 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 4037 . . . . 5  |-  ( U. U  C_  y  <->  A. x  e.  U  x  C_  y
)
3937, 38sylibr 204 . . . 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 958 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  y  e.  C )
4115mrcsscl 13837 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  y  /\  y  e.  C )  ->  ( F `  U. U ) 
C_  y )
4228, 39, 40, 41syl3anc 1184 . . 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 978 . . . 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 14576 . . . 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 1184 . . 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 224 . 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 14567 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  ( L `  U )  =  ( F `  U. U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   U.cuni 4007   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528  Moorecmre 13799  mrClscmrc 13800   Posetcpo 14389   lubclub 14391  toInccipo 14569
This theorem is referenced by:  mreclat  14605
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-13 1727  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  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  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-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-tset 13540  df-ple 13541  df-ocomp 13542  df-mre 13803  df-mrc 13804  df-poset 14395  df-lub 14423  df-ipo 14570
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