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Theorem mreclat 14306
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 14279 . . 3  |-  I  e. 
Poset
32a1i 10 . 2  |-  ( C  e.  (Moore `  X
)  ->  I  e.  Poset
)
4 eqid 2296 . . . . . . . 8  |-  (mrCls `  C )  =  (mrCls `  C )
5 eqid 2296 . . . . . . . 8  |-  ( lub `  I )  =  ( lub `  I )
61, 4, 5mrelatlub 14305 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  =  ( (mrCls `  C ) `  U. x ) )
7 uniss 3864 . . . . . . . . . 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 13518 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
109adantr 451 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. C  =  X )
118, 10sseqtrd 3227 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  U. x  C_  X )
124mrccl 13529 . . . . . . . 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 2370 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C )  ->  (
( lub `  I
) `  x )  e.  C )
15 fveq2 5541 . . . . . . . . . 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 2296 . . . . . . . . . . 11  |-  ( glb `  I )  =  ( glb `  I )
181, 17mrelatglb0 14304 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( ( glb `  I ) `  (/) )  =  X )
1918ad2antrr 706 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  (/) )  =  X )
2016, 19eqtrd 2328 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  =  X )
21 mre1cl 13512 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
2221ad2antrr 706 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  X  e.  C )
2320, 22eqeltrd 2370 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  x  C_  C )  /\  x  =  (/) )  ->  (
( glb `  I
) `  x )  e.  C )
241, 17mrelatglb 14303 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( glb `  I ) `
 x )  = 
|^| x )
25 mreintcl 13513 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
2624, 25eqeltrd 2370 . . . . . . . 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 2537 . . . . . 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 14274 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
32 sseq2 3213 . . . . . 6  |-  ( C  =  ( Base `  I
)  ->  ( x  C_  C  <->  x  C_  ( Base `  I ) ) )
33 eleq2 2357 . . . . . . 7  |-  ( C  =  ( Base `  I
)  ->  ( (
( lub `  I
) `  x )  e.  C  <->  ( ( lub `  I ) `  x
)  e.  ( Base `  I ) ) )
34 eleq2 2357 . . . . . . 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 1621 . 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 2296 . . 3  |-  ( Base `  I )  =  (
Base `  I )
4140, 5, 17isclat 14231 . 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 1530    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   U.cuni 3843   |^|cint 3878   ` cfv 5271   Basecbs 13164  Moorecmre 13500  mrClscmrc 13501   Posetcpo 14090   lubclub 14092   glbcglb 14093   CLatccla 14229  toInccipo 14270
This theorem is referenced by:  mreclatdemo  16849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-tset 13243  df-ple 13244  df-ocomp 13245  df-mre 13504  df-mrc 13505  df-poset 14096  df-lub 14124  df-glb 14125  df-clat 14230  df-odu 14249  df-ipo 14271
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