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

Step	Hyp	Ref	Expression
1		fvex 5882	. . . . 5 |- ( TopOpen ` W ) e. _V
2	1	uniex 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 ) ) )
4	2, 3	mp1i 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
6	5	sseli 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 )
9	7, 8	lssmre 17481	. . . . 5 |- ( W e. LMod -> ( LSubSp ` W ) e. (Moore ` ( Base ` W ) ) )
10	6, 9	syl 16	. . . 4 |- ( W e. ( TopSp i^i LMod ) -> ( LSubSp ` W ) e. (Moore ` ( Base ` W ) ) )
11		inss1 3723	. . . . . 6 |- ( TopSp i^i LMod ) C_ TopSp
12	11	sseli 3505	. . . . 5 |- ( W e. ( TopSp i^i LMod ) -> W e. TopSp )
13		eqid 2467	. . . . . . 7 |- ( TopOpen ` W ) = ( TopOpen ` W )
14	7, 13	tpsuni 19306	. . . . . 6 |- ( W e. TopSp -> ( Base ` W ) = U. ( TopOpen ` W ) )
15	14	fveq2d 5876	. . . . 5 |- ( W e. TopSp -> (Moore ` ( Base ` W ) ) = (Moore ` U. ( TopOpen ` W ) ) )
16	12, 15	syl 16	. . . 4 |- ( W e. ( TopSp i^i LMod ) -> (Moore ` ( Base ` W ) ) = (Moore ` U. ( TopOpen ` W ) ) )
17	10, 16	eleqtrd 2557	. . 3 |- ( W e. ( TopSp i^i LMod ) -> ( LSubSp ` W ) e. (Moore ` U. ( TopOpen ` W ) ) )
18	13	tpstop 19307	. . . 4 |- ( W e. TopSp -> ( TopOpen ` W ) e. Top )
19		eqid 2467	. . . . 5 |- U. ( TopOpen ` W ) = U. ( TopOpen ` W )
20	19	cldmre 19445	. . . 4 |- ( ( TopOpen ` W ) e. Top -> ( Clsd ` ( TopOpen ` W ) ) e. (Moore ` U. ( TopOpen ` W ) ) )
21	12, 18, 20	3syl 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 ) ) )
23	4, 17, 21, 22	syl3anc 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 )
25	23, 24	syl 16	1 |- ( W e. ( TopSp i^i LMod ) -> (toInc ` ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) ) e. CLat )

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)