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Theorem mrelatlub 14289
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 2283 . 2  |-  ( le
`  I )  =  ( le `  I
)
2 mreclat.i . . . 4  |-  I  =  (toInc `  C )
32ipobas 14258 . . 3  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
43adantr 451 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  C  =  ( Base `  I
) )
5 mrelatlub.l . . 3  |-  L  =  ( lub `  I
)
65a1i 10 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  L  =  ( lub `  I
) )
72ipopos 14263 . . 3  |-  I  e. 
Poset
87a1i 10 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  I  e.  Poset )
9 simpr 447 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U  C_  C )
10 uniss 3848 . . . . 5  |-  ( U 
C_  C  ->  U. U  C_ 
U. C )
1110adantl 452 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_ 
U. C )
12 mreuni 13502 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
1312adantr 451 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. C  =  X )
1411, 13sseqtrd 3214 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_  X )
15 mrelatlub.f . . . 4  |-  F  =  (mrCls `  C )
1615mrccl 13513 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  -> 
( F `  U. U )  e.  C
)
1714, 16syldan 456 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  ( F `  U. U )  e.  C )
18 elssuni 3855 . . . 4  |-  ( x  e.  U  ->  x  C_ 
U. U )
1915mrcssid 13519 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  X )  ->  U. U  C_  ( F `
 U. U ) )
2014, 19syldan 456 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  U. U  C_  ( F `  U. U ) )
2118, 20sylan9ssr 3193 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  x  C_  ( F `  U. U ) )
22 simpll 730 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
239sselda 3180 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  x  e.  C )
2417adantr 451 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  ( F `  U. U )  e.  C )
252, 1ipole 14261 . . . 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 1182 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  (
x ( le `  I ) ( F `
 U. U )  <-> 
x  C_  ( F `  U. U ) ) )
2721, 26mpbird 223 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  x  e.  U )  ->  x
( le `  I
) ( F `  U. U ) )
28 simp1l 979 . . . 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 734 . . . . . . . . 9  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
30 simplr 731 . . . . . . . . . 10  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C )  ->  U  C_  C )
3130sselda 3180 . . . . . . . . 9  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  x  e.  C )
32 simplr 731 . . . . . . . . 9  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  y  e.  C )
332, 1ipole 14261 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  C  /\  y  e.  C )  ->  (
x ( le `  I ) y  <->  x  C_  y
) )
3429, 31, 32, 33syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  (
x ( le `  I ) y  <->  x  C_  y
) )
3534biimpd 198 . . . . . . 7  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C
)  /\  x  e.  U )  ->  (
x ( le `  I ) y  ->  x  C_  y ) )
3635ralimdva 2621 . . . . . 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 1148 . . . . 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 3857 . . . . 5  |-  ( U. U  C_  y  <->  A. x  e.  U  x  C_  y
)
3937, 38sylibr 203 . . . 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 956 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C )  /\  y  e.  C  /\  A. x  e.  U  x ( le `  I ) y )  ->  y  e.  C )
4115mrcsscl 13522 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U. U  C_  y  /\  y  e.  C )  ->  ( F `  U. U ) 
C_  y )
4228, 39, 40, 41syl3anc 1182 . . 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 976 . . . 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 14261 . . . 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 1182 . . 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 223 . 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 14252 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  ( L `  U )  =  ( F `  U. U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   U.cuni 3827   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215  Moorecmre 13484  mrClscmrc 13485   Posetcpo 14074   lubclub 14076  toInccipo 14254
This theorem is referenced by:  mreclat  14290
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-13 1686  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  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-tset 13227  df-ple 13228  df-ocomp 13229  df-mre 13488  df-mrc 13489  df-poset 14080  df-lub 14108  df-ipo 14255
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