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Theorem mrelatlub 14567
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 2404 . 2  |-  ( le
`  I )  =  ( le `  I
)
2 mreclat.i . . . 4  |-  I  =  (toInc `  C )
32ipobas 14536 . . 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 14541 . . 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 3996 . . . . 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 13780 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
1312adantr 452 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. C  =  X )
1411, 13sseqtrd 3344 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_  X )
15 mrelatlub.f . . . 4  |-  F  =  (mrCls `  C )
1615mrccl 13791 . . 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 4003 . . . 4  |-  ( x  e.  U  ->  x  C_ 
U. U )
1915mrcssid 13797 . . . . 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 3322 . . 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 3308 . . . 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 14539 . . . 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 3308 . . . . . . . . 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 14539 . . . . . . . . 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 2744 . . . . . 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 4005 . . . . 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 13800 . . . 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 14539 . . . 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 14530 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 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   U.cuni 3975   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491  Moorecmre 13762  mrClscmrc 13763   Posetcpo 14352   lubclub 14354  toInccipo 14532
This theorem is referenced by:  mreclat  14568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-tset 13503  df-ple 13504  df-ocomp 13505  df-mre 13766  df-mrc 13767  df-poset 14358  df-lub 14386  df-ipo 14533
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