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Theorem mreclat 14533
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 14506 . . 3  |-  I  e. 
Poset
32a1i 11 . 2  |-  ( C  e.  (Moore `  X
)  ->  I  e.  Poset
)
4 eqid 2380 . . . . . . . 8  |-  (mrCls `  C )  =  (mrCls `  C )
5 eqid 2380 . . . . . . . 8  |-  ( lub `  I )  =  ( lub `  I )
61, 4, 5mrelatlub 14532 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  =  ( (mrCls `  C ) `  U. x ) )
7 uniss 3971 . . . . . . . . . 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 13745 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
109adantr 452 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. C  =  X )
118, 10sseqtrd 3320 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. x  C_  X )
124mrccl 13756 . . . . . . . 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 2454 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  e.  C )
15 fveq2 5661 . . . . . . . . . 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 2380 . . . . . . . . . . 11  |-  ( glb `  I )  =  ( glb `  I )
181, 17mrelatglb0 14531 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( ( glb `  I ) `  (/) )  =  X )
1918ad2antrr 707 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  (/) )  =  X )
2016, 19eqtrd 2412 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  =  X )
21 mre1cl 13739 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
2221ad2antrr 707 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  X  e.  C )
2320, 22eqeltrd 2454 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  e.  C )
241, 17mrelatglb 14530 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( glb `  I ) `
 x )  = 
|^| x )
25 mreintcl 13740 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
2624, 25eqeltrd 2454 . . . . . . . 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 2621 . . . . . 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 14501 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
32 sseq2 3306 . . . . . 6  |-  ( C  =  ( Base `  I
)  ->  ( x  C_  C  <->  x  C_  ( Base `  I ) ) )
33 eleq2 2441 . . . . . . 7  |-  ( C  =  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  C  <->  ( ( lub `  I ) `  x
)  e.  ( Base `  I ) ) )
34 eleq2 2441 . . . . . . 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 1638 . 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 2380 . . 3  |-  ( Base `  I )  =  (
Base `  I )
4140, 5, 17isclat 14458 . 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 1546    = wceq 1649    e. wcel 1717    =/= wne 2543    C_ wss 3256   (/)c0 3564   U.cuni 3950   |^|cint 3985   ` cfv 5387   Basecbs 13389  Moorecmre 13727  mrClscmrc 13728   Posetcpo 14317   lubclub 14319   glbcglb 14320   CLatccla 14456  toInccipo 14497
This theorem is referenced by:  mreclatdemo  17076
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-tset 13468  df-ple 13469  df-ocomp 13470  df-mre 13731  df-mrc 13732  df-poset 14323  df-lub 14351  df-glb 14352  df-clat 14457  df-odu 14476  df-ipo 14498
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