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Theorem mreclat 14605
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 14578 . . 3  |-  I  e. 
Poset
32a1i 11 . 2  |-  ( C  e.  (Moore `  X
)  ->  I  e.  Poset
)
4 eqid 2435 . . . . . . . 8  |-  (mrCls `  C )  =  (mrCls `  C )
5 eqid 2435 . . . . . . . 8  |-  ( lub `  I )  =  ( lub `  I )
61, 4, 5mrelatlub 14604 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  =  ( (mrCls `  C ) `  U. x ) )
7 uniss 4028 . . . . . . . . . 10  |-  ( x 
C_  C  ->  U. x  C_ 
U. C )
87adantl 453 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. x  C_ 
U. C )
9 mreuni 13817 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
109adantr 452 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. C  =  X )
118, 10sseqtrd 3376 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. x  C_  X )
124mrccl 13828 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  U. x  C_  X )  -> 
( (mrCls `  C
) `  U. x )  e.  C )
1311, 12syldan 457 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
(mrCls `  C ) `  U. x )  e.  C )
146, 13eqeltrd 2509 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  e.  C )
15 fveq2 5720 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( ( glb `  I ) `
 x )  =  ( ( glb `  I
) `  (/) ) )
1615adantl 453 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  =  ( ( glb `  I ) `  (/) ) )
17 eqid 2435 . . . . . . . . . . 11  |-  ( glb `  I )  =  ( glb `  I )
181, 17mrelatglb0 14603 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( ( glb `  I ) `  (/) )  =  X )
1918ad2antrr 707 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  (/) )  =  X )
2016, 19eqtrd 2467 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  =  X )
21 mre1cl 13811 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
2221ad2antrr 707 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  X  e.  C )
2320, 22eqeltrd 2509 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  e.  C )
241, 17mrelatglb 14602 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( glb `  I ) `
 x )  = 
|^| x )
25 mreintcl 13812 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
2624, 25eqeltrd 2509 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( glb `  I ) `
 x )  e.  C )
27263expa 1153 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =/=  (/) )  ->  (
( glb `  I
) `  x )  e.  C )
2823, 27pm2.61dane 2676 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( glb `  I
) `  x )  e.  C )
2914, 28jca 519 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( ( lub `  I
) `  x )  e.  C  /\  (
( glb `  I
) `  x )  e.  C ) )
3029ex 424 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( x  C_  C  ->  ( (
( lub `  I
) `  x )  e.  C  /\  (
( glb `  I
) `  x )  e.  C ) ) )
311ipobas 14573 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
32 sseq2 3362 . . . . . 6  |-  ( C  =  ( Base `  I
)  ->  ( x  C_  C  <->  x  C_  ( Base `  I ) ) )
33 eleq2 2496 . . . . . . 7  |-  ( C  =  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  C  <->  ( ( lub `  I ) `  x
)  e.  ( Base `  I ) ) )
34 eleq2 2496 . . . . . . 7  |-  ( C  =  ( Base `  I
)  ->  ( (
( glb `  I
) `  x )  e.  C  <->  ( ( glb `  I ) `  x
)  e.  ( Base `  I ) ) )
3533, 34anbi12d 692 . . . . . 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 312 . . . . 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 16 . . . 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 202 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( x  C_  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  ( Base `  I
)  /\  ( ( glb `  I ) `  x )  e.  (
Base `  I )
) ) )
3938alrimiv 1641 . 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 2435 . . 3  |-  ( Base `  I )  =  (
Base `  I )
4140, 5, 17isclat 14530 . 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 646 1  |-  ( C  e.  (Moore `  X
)  ->  I  e.  CLat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   U.cuni 4007   |^|cint 4042   ` cfv 5446   Basecbs 13461  Moorecmre 13799  mrClscmrc 13800   Posetcpo 14389   lubclub 14391   glbcglb 14392   CLatccla 14528  toInccipo 14569
This theorem is referenced by:  mreclatdemo  17152
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-sets 13467  df-tset 13540  df-ple 13541  df-ocomp 13542  df-mre 13803  df-mrc 13804  df-poset 14395  df-lub 14423  df-glb 14424  df-clat 14529  df-odu 14548  df-ipo 14570
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