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Theorem mreclatdemoBAD 19463
Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 15690. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6256 update): This proof uses the old df-clat 15611 and references the required instance of mreclatBAD 15690 as a hypothesis. When mreclatBAD 15690 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below.
Hypothesis
Ref Expression
mreclatBAD.  |-  ( ( ( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen
`  W ) ) )  e.  (Moore `  U. ( TopOpen `  W )
)  ->  (toInc `  (
( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen
`  W ) ) ) )  e.  CLat )
Assertion
Ref Expression
mreclatdemoBAD  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  (toInc `  (
( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen
`  W ) ) ) )  e.  CLat )

Proof of Theorem mreclatdemoBAD
StepHypRef Expression
1 fvex 5882 . . . . 5  |-  ( TopOpen `  W )  e.  _V
21uniex 6591 . . . 4  |-  U. ( TopOpen
`  W )  e. 
_V
3 mremre 14875 . . . 4  |-  ( U. ( TopOpen `  W )  e.  _V  ->  (Moore `  U. ( TopOpen `  W )
)  e.  (Moore `  ~P U. ( TopOpen `  W
) ) )
42, 3mp1i 12 . . 3  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  (Moore `  U. ( TopOpen `  W )
)  e.  (Moore `  ~P U. ( TopOpen `  W
) ) )
5 inss2 3724 . . . . . 6  |-  ( TopSp  i^i 
LMod )  C_  LMod
65sseli 3505 . . . . 5  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  W  e.  LMod )
7 eqid 2467 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
8 eqid 2467 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
97, 8lssmre 17481 . . . . 5  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  e.  (Moore `  ( Base `  W
) ) )
106, 9syl 16 . . . 4  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  ( LSubSp `  W
)  e.  (Moore `  ( Base `  W )
) )
11 inss1 3723 . . . . . 6  |-  ( TopSp  i^i 
LMod )  C_  TopSp
1211sseli 3505 . . . . 5  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  W  e.  TopSp )
13 eqid 2467 . . . . . . 7  |-  ( TopOpen `  W )  =  (
TopOpen `  W )
147, 13tpsuni 19306 . . . . . 6  |-  ( W  e.  TopSp  ->  ( Base `  W )  =  U. ( TopOpen `  W )
)
1514fveq2d 5876 . . . . 5  |-  ( W  e.  TopSp  ->  (Moore `  ( Base `  W ) )  =  (Moore `  U. ( TopOpen `  W )
) )
1612, 15syl 16 . . . 4  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  (Moore `  ( Base `  W ) )  =  (Moore `  U. ( TopOpen `  W )
) )
1710, 16eleqtrd 2557 . . 3  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  ( LSubSp `  W
)  e.  (Moore `  U. ( TopOpen `  W )
) )
1813tpstop 19307 . . . 4  |-  ( W  e.  TopSp  ->  ( TopOpen `  W )  e.  Top )
19 eqid 2467 . . . . 5  |-  U. ( TopOpen
`  W )  = 
U. ( TopOpen `  W
)
2019cldmre 19445 . . . 4  |-  ( (
TopOpen `  W )  e. 
Top  ->  ( Clsd `  ( TopOpen
`  W ) )  e.  (Moore `  U. ( TopOpen `  W )
) )
2112, 18, 203syl 20 . . 3  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  ( Clsd `  ( TopOpen
`  W ) )  e.  (Moore `  U. ( TopOpen `  W )
) )
22 mreincl 14870 . . 3  |-  ( ( (Moore `  U. ( TopOpen `  W ) )  e.  (Moore `  ~P U. ( TopOpen
`  W ) )  /\  ( LSubSp `  W
)  e.  (Moore `  U. ( TopOpen `  W )
)  /\  ( Clsd `  ( TopOpen `  W )
)  e.  (Moore `  U. ( TopOpen `  W )
) )  ->  (
( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen
`  W ) ) )  e.  (Moore `  U. ( TopOpen `  W )
) )
234, 17, 21, 22syl3anc 1228 . 2  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  ( ( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen `  W
) ) )  e.  (Moore `  U. ( TopOpen `  W ) ) )
24 mreclatBAD. . 2  |-  ( ( ( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen
`  W ) ) )  e.  (Moore `  U. ( TopOpen `  W )
)  ->  (toInc `  (
( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen
`  W ) ) ) )  e.  CLat )
2523, 24syl 16 1  |-  ( W  e.  ( TopSp  i^i  LMod )  ->  (toInc `  (
( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen
`  W ) ) ) )  e.  CLat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480   ~Pcpw 4016   U.cuni 4251   ` cfv 5594   Basecbs 14506   TopOpenctopn 14693  Moorecmre 14853   CLatccla 15610  toInccipo 15654   LModclmod 17381   LSubSpclss 17447   Topctop 19261   TopSpctps 19264   Clsdccld 19383
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-iin 4334  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-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  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-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-plusg 14584  df-0g 14713  df-mre 14857  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mgp 17012  df-ur 17024  df-ring 17070  df-lmod 17383  df-lss 17448  df-top 19266  df-topon 19269  df-topsp 19270  df-cld 19386
This theorem is referenced by: (None)
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