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Theorem mreclat 14290
Description: A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mreclat.i  |-  I  =  (toInc `  C )
Assertion
Ref Expression
mreclat  |-  ( C  e.  (Moore `  X
)  ->  I  e.  CLat )

Proof of Theorem mreclat
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mreclat.i . . . 4  |-  I  =  (toInc `  C )
21ipopos 14263 . . 3  |-  I  e. 
Poset
32a1i 10 . 2  |-  ( C  e.  (Moore `  X
)  ->  I  e.  Poset
)
4 eqid 2283 . . . . . . . 8  |-  (mrCls `  C )  =  (mrCls `  C )
5 eqid 2283 . . . . . . . 8  |-  ( lub `  I )  =  ( lub `  I )
61, 4, 5mrelatlub 14289 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  =  ( (mrCls `  C ) `  U. x ) )
7 uniss 3848 . . . . . . . . . 10  |-  ( x 
C_  C  ->  U. x  C_ 
U. C )
87adantl 452 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. x  C_ 
U. C )
9 mreuni 13502 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
109adantr 451 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. C  =  X )
118, 10sseqtrd 3214 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. x  C_  X )
124mrccl 13513 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  U. x  C_  X )  -> 
( (mrCls `  C
) `  U. x )  e.  C )
1311, 12syldan 456 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
(mrCls `  C ) `  U. x )  e.  C )
146, 13eqeltrd 2357 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  e.  C )
15 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( ( glb `  I ) `
 x )  =  ( ( glb `  I
) `  (/) ) )
1615adantl 452 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  =  ( ( glb `  I ) `  (/) ) )
17 eqid 2283 . . . . . . . . . . 11  |-  ( glb `  I )  =  ( glb `  I )
181, 17mrelatglb0 14288 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( ( glb `  I ) `  (/) )  =  X )
1918ad2antrr 706 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  (/) )  =  X )
2016, 19eqtrd 2315 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  =  X )
21 mre1cl 13496 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
2221ad2antrr 706 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  X  e.  C )
2320, 22eqeltrd 2357 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  e.  C )
241, 17mrelatglb 14287 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( glb `  I ) `
 x )  = 
|^| x )
25 mreintcl 13497 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
2624, 25eqeltrd 2357 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( glb `  I ) `
 x )  e.  C )
27263expa 1151 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =/=  (/) )  ->  (
( glb `  I
) `  x )  e.  C )
2823, 27pm2.61dane 2524 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( glb `  I
) `  x )  e.  C )
2914, 28jca 518 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( ( lub `  I
) `  x )  e.  C  /\  (
( glb `  I
) `  x )  e.  C ) )
3029ex 423 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( x  C_  C  ->  ( (
( lub `  I
) `  x )  e.  C  /\  (
( glb `  I
) `  x )  e.  C ) ) )
311ipobas 14258 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
32 sseq2 3200 . . . . . 6  |-  ( C  =  ( Base `  I
)  ->  ( x  C_  C  <->  x  C_  ( Base `  I ) ) )
33 eleq2 2344 . . . . . . 7  |-  ( C  =  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  C  <->  ( ( lub `  I ) `  x
)  e.  ( Base `  I ) ) )
34 eleq2 2344 . . . . . . 7  |-  ( C  =  ( Base `  I
)  ->  ( (
( glb `  I
) `  x )  e.  C  <->  ( ( glb `  I ) `  x
)  e.  ( Base `  I ) ) )
3533, 34anbi12d 691 . . . . . 6  |-  ( C  =  ( Base `  I
)  ->  ( (
( ( lub `  I
) `  x )  e.  C  /\  (
( glb `  I
) `  x )  e.  C )  <->  ( (
( lub `  I
) `  x )  e.  ( Base `  I
)  /\  ( ( glb `  I ) `  x )  e.  (
Base `  I )
) ) )
3632, 35imbi12d 311 . . . . 5  |-  ( C  =  ( Base `  I
)  ->  ( (
x  C_  C  ->  ( ( ( lub `  I
) `  x )  e.  C  /\  (
( glb `  I
) `  x )  e.  C ) )  <->  ( x  C_  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  ( Base `  I
)  /\  ( ( glb `  I ) `  x )  e.  (
Base `  I )
) ) ) )
3731, 36syl 15 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( (
x  C_  C  ->  ( ( ( lub `  I
) `  x )  e.  C  /\  (
( glb `  I
) `  x )  e.  C ) )  <->  ( x  C_  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  ( Base `  I
)  /\  ( ( glb `  I ) `  x )  e.  (
Base `  I )
) ) ) )
3830, 37mpbid 201 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( x  C_  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  ( Base `  I
)  /\  ( ( glb `  I ) `  x )  e.  (
Base `  I )
) ) )
3938alrimiv 1617 . 2  |-  ( C  e.  (Moore `  X
)  ->  A. x
( x  C_  ( Base `  I )  -> 
( ( ( lub `  I ) `  x
)  e.  ( Base `  I )  /\  (
( glb `  I
) `  x )  e.  ( Base `  I
) ) ) )
40 eqid 2283 . . 3  |-  ( Base `  I )  =  (
Base `  I )
4140, 5, 17isclat 14215 . 2  |-  ( I  e.  CLat  <->  ( I  e. 
Poset  /\  A. x ( x  C_  ( Base `  I )  ->  (
( ( lub `  I
) `  x )  e.  ( Base `  I
)  /\  ( ( glb `  I ) `  x )  e.  (
Base `  I )
) ) ) )
423, 39, 41sylanbrc 645 1  |-  ( C  e.  (Moore `  X
)  ->  I  e.  CLat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   U.cuni 3827   |^|cint 3862   ` cfv 5255   Basecbs 13148  Moorecmre 13484  mrClscmrc 13485   Posetcpo 14074   lubclub 14076   glbcglb 14077   CLatccla 14213  toInccipo 14254
This theorem is referenced by:  mreclatdemo  16833
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-sets 13154  df-tset 13227  df-ple 13228  df-ocomp 13229  df-mre 13488  df-mrc 13489  df-poset 14080  df-lub 14108  df-glb 14109  df-clat 14214  df-odu 14233  df-ipo 14255
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