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Theorem mreclatBAD 15691
Description: A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6256 update): Reprove using isclat 15613 instead of the isclatBAD. hypothesis. See commented-out mreclat above.
Hypotheses
Ref Expression
mreclat.i  |-  I  =  (toInc `  C )
isclatBAD.  |-  ( I  e.  CLat  <->  ( I  e. 
Poset  /\  A. x ( x  C_  ( Base `  I )  ->  (
( ( lub `  I
) `  x )  e.  ( Base `  I
)  /\  ( ( glb `  I ) `  x )  e.  (
Base `  I )
) ) ) )
Assertion
Ref Expression
mreclatBAD  |-  ( C  e.  (Moore `  X
)  ->  I  e.  CLat )
Distinct variable groups:    x, I    x, C    x, X

Proof of Theorem mreclatBAD
StepHypRef Expression
1 mreclat.i . . . 4  |-  I  =  (toInc `  C )
21ipopos 15664 . . 3  |-  I  e. 
Poset
32a1i 11 . 2  |-  ( C  e.  (Moore `  X
)  ->  I  e.  Poset
)
4 eqid 2467 . . . . . . . 8  |-  (mrCls `  C )  =  (mrCls `  C )
5 eqid 2467 . . . . . . . 8  |-  ( lub `  I )  =  ( lub `  I )
61, 4, 5mrelatlub 15690 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  =  ( (mrCls `  C ) `  U. x ) )
7 uniss 4272 . . . . . . . . . 10  |-  ( x 
C_  C  ->  U. x  C_ 
U. C )
87adantl 466 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. x  C_ 
U. C )
9 mreuni 14872 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
109adantr 465 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. C  =  X )
118, 10sseqtrd 3545 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. x  C_  X )
124mrccl 14883 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  U. x  C_  X )  -> 
( (mrCls `  C
) `  U. x )  e.  C )
1311, 12syldan 470 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
(mrCls `  C ) `  U. x )  e.  C )
146, 13eqeltrd 2555 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  e.  C )
15 fveq2 5872 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( ( glb `  I ) `
 x )  =  ( ( glb `  I
) `  (/) ) )
1615adantl 466 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  =  ( ( glb `  I ) `  (/) ) )
17 eqid 2467 . . . . . . . . . . 11  |-  ( glb `  I )  =  ( glb `  I )
181, 17mrelatglb0 15689 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( ( glb `  I ) `  (/) )  =  X )
1918ad2antrr 725 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  (/) )  =  X )
2016, 19eqtrd 2508 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  =  X )
21 mre1cl 14866 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
2221ad2antrr 725 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  X  e.  C )
2320, 22eqeltrd 2555 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  e.  C )
241, 17mrelatglb 15688 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( glb `  I ) `
 x )  = 
|^| x )
25 mreintcl 14867 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
2624, 25eqeltrd 2555 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( glb `  I ) `
 x )  e.  C )
27263expa 1196 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =/=  (/) )  ->  (
( glb `  I
) `  x )  e.  C )
2823, 27pm2.61dane 2785 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( glb `  I
) `  x )  e.  C )
2914, 28jca 532 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( ( lub `  I
) `  x )  e.  C  /\  (
( glb `  I
) `  x )  e.  C ) )
3029ex 434 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( x  C_  C  ->  ( (
( lub `  I
) `  x )  e.  C  /\  (
( glb `  I
) `  x )  e.  C ) ) )
311ipobas 15659 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
32 sseq2 3531 . . . . . 6  |-  ( C  =  ( Base `  I
)  ->  ( x  C_  C  <->  x  C_  ( Base `  I ) ) )
33 eleq2 2540 . . . . . . 7  |-  ( C  =  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  C  <->  ( ( lub `  I ) `  x
)  e.  ( Base `  I ) ) )
34 eleq2 2540 . . . . . . 7  |-  ( C  =  ( Base `  I
)  ->  ( (
( glb `  I
) `  x )  e.  C  <->  ( ( glb `  I ) `  x
)  e.  ( Base `  I ) ) )
3533, 34anbi12d 710 . . . . . 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 320 . . . . 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 210 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( x  C_  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  ( Base `  I
)  /\  ( ( glb `  I ) `  x )  e.  (
Base `  I )
) ) )
3938alrimiv 1695 . 2  |-  ( C  e.  (Moore `  X
)  ->  A. x
( x  C_  ( Base `  I )  -> 
( ( ( lub `  I ) `  x
)  e.  ( Base `  I )  /\  (
( glb `  I
) `  x )  e.  ( Base `  I
) ) ) )
40 isclatBAD. . 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 )
) ) ) )
413, 39, 40sylanbrc 664 1  |-  ( C  e.  (Moore `  X
)  ->  I  e.  CLat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3481   (/)c0 3790   U.cuni 4251   |^|cint 4288   ` cfv 5594   Basecbs 14507  Moorecmre 14854  mrClscmrc 14855   Posetcpo 15444   lubclub 15446   glbcglb 15447   CLatccla 15611  toInccipo 15655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-tset 14591  df-ple 14592  df-ocomp 14593  df-mre 14858  df-mrc 14859  df-poset 15450  df-lub 15478  df-glb 15479  df-odu 15633  df-ipo 15656
This theorem is referenced by: (None)
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