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Theorem mrelatlub 15685
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 2441 . 2  |-  ( le
`  I )  =  ( le `  I
)
2 mreclat.i . . . 4  |-  I  =  (toInc `  C )
32ipobas 15654 . . 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 15659 . . 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 4251 . . . . 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 14869 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
1312adantr 465 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. C  =  X )
1411, 13sseqtrd 3522 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_  X )
15 mrelatlub.f . . . 4  |-  F  =  (mrCls `  C )
1615mrccl 14880 . . 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 4260 . . . 4  |-  ( x  e.  U  ->  x  C_ 
U. U )
1915mrcssid 14886 . . . . 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 3500 . . 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 3486 . . . 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 15657 . . . 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 1227 . . 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 1019 . . . 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 3486 . . . . . . . . 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 15657 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  C  /\  y  e.  C )  ->  (
x ( le `  I ) y  <->  x  C_  y
) )
3429, 31, 32, 33syl3anc 1227 . . . . . . . 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 2849 . . . . . 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 1192 . . . . 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 4262 . . . . 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 996 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  y  e.  C )
4115mrcsscl 14889 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  y  /\  y  e.  C )  ->  ( F `  U. U ) 
C_  y )
4228, 39, 40, 41syl3anc 1227 . . 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 1016 . . . 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 15657 . . . 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 1227 . . 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 15648 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 972    = wceq 1381    e. wcel 1802   A.wral 2791    C_ wss 3458   U.cuni 4230   class class class wbr 4433   ` cfv 5574   Basecbs 14504   lecple 14576  Moorecmre 14851  mrClscmrc 14852   Posetcpo 15438   lubclub 15440  toInccipo 15650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-fz 11677  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-tset 14588  df-ple 14589  df-ocomp 14590  df-mre 14855  df-mrc 14856  df-preset 15426  df-poset 15444  df-lub 15473  df-ipo 15651
This theorem is referenced by:  mreclatBAD  15686
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